Manfred Knebusch

Specialization of Quadratic and Symmetric Bilinear Forms

Translated from German by Thomas Unger

Preliminary Version (22 Nov 2009)

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Dedicated to the memory of my teachers

Emil Artin 1898 – 1962 Hel Braun 1914 – 1986 Ernst Witt 1911 – 1991

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Preface

A Mathematician Said Who

Can Quote Me a Theorem that’s True? For the ones that I Know Are Simply not So, When the Characteristic is Two!

This pretty limerick first came to my ears in May 1998 during a talk by T.Y. Lam on field invariants from the theory of quadratic forms.^{1} It is – poetic exaggeration allowed – a suitable motto for this monograph.

What is it about? In the beginning of the seventies I drew up a specialization theory of quadratic and symmetric bilinear forms over fields [K_{4}]. Let λ: K → L ∪ ∞ be a place. Then one can assign a form λ_{∗}(φ) to a form φ over K in a meaningful way if φ has “good reduction” with respect to λ (see §1). The basic idea is to simply apply the place λ to the coefficients of φ which therefore of course have to be in the valuation ring of λ.

The specialization theory of that time was satisfactory as long as the field L, and therefore also K, had characteristic ≠ 2. It served me in the first place as the foundation for a theory of generic splitting of quadratic forms [K_{5}], [K_{6}]. After a very modest beginning, this theory is now in full bloom. It became important for the understanding of quadratic forms over fields, as can be seen from the book [IKKV] of Izhboldin-Kahn-Karpenko-Vishik for instance. One should note that there exists a theory of (partial) generic splitting of central simple algebras and reductive algebraic groups, parallel to the theory of generic splitting of quadratic forms (see [Ke R] and the literature cited there).

In this book I would like to present a specialization theory of quadratic and symmetric bilinear forms with respect to a place λ: K → L ∪ ∞, without the assumption that char L ≠ 2. This is where complications arise. We have to make a distinction between bilinear and quadratic forms and study them both over fields and valuation rings. From the viewpoint of reductive algebraic groups, the so-called regular quadratic forms (see below) are the natural objects. But, even if we are only interested in such forms, we have to know a bit about specialization of nondegenerate symmetric bilinear forms, since

1 “Some reflections on quadratic invariants of fields”, 3 May 1998 in Notre Dame (Indiana) on the occasion of O.T. O’Meara’s 70th birthday.

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VIII Specialization of Quadratic and Symmetric Bilinear Forms

they occur as “multipliers” of quadratic forms: if φ is such a bilinear form and ψ is a regular quadratic form, then we can form a tensor product φ ⊗ ψ, see §5. This is a quadratic form, which is again regular when ψ has even dimension (dim ψ = number of variables occurring in ψ). However – and here already we run into trouble – when dim ψ is odd, φ ⊗ ψ is not necessarily regular.

Even if we only want to understand quadratic forms over a field K of characteristic zero, it might be necessary to look at specializations with respect to places from K to fields of characteristic 2, especially in arithmetic investigations. When K itself has characteristic 2, an often more complicated situation may occur, for which we are not prepared by the available literature. Surely, fields of characteristic 2 were already allowed in my work on specializations in 1973 [K_{4}], but from today’s point of view satisfactory results were only obtained for symmetric bilinear forms. For quadratic forms there are gaping holes. We have to study quadratic forms over a valuation ring in which 2 is not a unit. Even the beautiful and extensive book of Ricardo Baeza [Ba] doesn’t give us enough for the theory of specializations, although Baeza even allows semilocal rings instead of valuation rings. He only studies quadratic forms whose associated bilinear forms are nondegenerate. This forces those forms to have even dimension.

Let me now discuss the contents of this book. After an introduction to the problem in §1, which can be understood without any previous knowledge of quadratic and bilinear forms, the specialization theory of symmetric bilinear forms is presented in §2 - §3. There are good, generally accessible sources available for the foundations of the algebraic theory of symmetric bilinear forms. Therefore many results are presented without a proof, but with a reference to the literature instead. As an important application, the outlines of the theory of generic splitting in characteristic ≠ 2 are sketched in §4, nearly without proofs.

From §5 onwards we address the theory of quadratic forms. In characteristic 2 fewer results can be found in the literature for such forms than for bilinear forms, even at the basic level. Therefore we present most of the proofs. We also concern ourselves with the so-called “weak specialization” (see §1) and get into areas which may seem strange even to specialists in the theory of quadratic forms. In particular we have to require a quadratic form over K to be “obedient” in order to weakly specialize it with respect to a place λ: K → L∪∞ (see §7). I have never encountered such a thing anywhere in the literature.

At the end of Chapter I we reach a level in the specialization theory of quadratic forms that facilitates a generic splitting theory, useful for many applications. In the first two sections (§9, §10) of Chapter II we produce such a generic splitting theory in two versions, both of which deserve interest in their own right.

We call a quadratic form φ over a field k nondegenerate when its quasilinear part (cf. Arf [A]), which we denote by QL(φ), is anisotropic. We further call – deviating from Arf [A] – φ regular when QL(φ) is at most onedimensional and strictly regular when QL(φ) = 0 (cf. §6, Definition 3). When

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k has characteristic ≠ 2, every nondegenerate form is strictly regular, but in characteristic 2 the quasilinear part causes complications. For in this case φ can become degenerate under a field extension L ⊃ k. Only in the regular case is this impossible.

In §9 we study the splitting behaviour of a regular quadratic form φ over k under field extensions, while in §10 any nondegenerate form φ, but only separable extensions of k are allowed. The theory of §9 incorporates the theory of §4, and so the missing proofs of §4 are subsequently filled in.

Until the end of §10 our specialization theory is based on an obvious “canonical” concept of good reduction of a form φ over a field K (quadratic or symmetric bilinear) to a valuation ring 𝔬 of K, similar to what is known under this name in other areas of mathematics (e.g. abelian varieties). There is nothing wrong with this theory, however for many applications it is too limited.

This is particularly clear when studying specializations with respect to a place λ: K → L ∪ ∞ with char K = 0, char L = 2. If φ is a nondegenerate quadratic form over K with good reduction with respect to λ, then the specialization λ_{∗}(φ) is automatically strictly regular. However, we would like to have a more general specialization concept, in which forms with quasilinear part ≠ 0 can arise over L. Conversely, if the place λ is surjective, i.e. λ(K) = L ∪ ∞, we would like to “lift” every nondegenerate quadratic form ψ over L with respect to λ to a form φ over K, i.e. to find a form φ over K which specializes to ψ with respect to λ. Then we could use the theory of forms over K to make statements about ψ.

We present such a general specialization theory in §11. It is based on the concept of “fair reduction”, which is less orthodox than good reduction, but which possesses nevertheless quite satisfying properties.

Next, in §12, we present a theory of generic splitting, which unites the theories of §4, §9 and §10 under one roof and which incorporates fair reduction. This theory is deepened in §13 and §14 through the study of generic splitting towers and so we reach the end of Chapter II.

Chapter III (§15 - §27) is a long chapter in which we present a panorama of results about quadratic forms over fields for which specialization and generic splitting of forms play an important role. This only scratches the surface of applications of the specialization theory of Chapters I and II. Certainly many more results can be unearthed.

We return to the foundations of specialization theory in the final short Chapter IV (§28 - §32). Quadratic and bilinear forms over a field can be specialized with respect to a more general “quadratic place” Λ: K → L ∪ ∞ (defined in §28) instead of a usual place λ: K → L ∪ ∞. This represents a considerable broadening of the specialization theory of Chapters I and II. Of course we require again “obedience” from a quadratic form q over K in order for its specialization Λ_{∗}(q) to reasonably exist. It then turns out that the generic splitting behaviour of Λ_{∗}(q) is governed by the splitting behaviour of q and Λ, in so far good or fair reduction is present in a weak sense, as elucidated for ordinary places in Chapter II.

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Why are quadratic places of interest, compared to ordinary places? To answer this question we observe the following. If a form q over K has bad reduction with respect to a place λ: K → L ∪ ∞ , it often happens that λ can be “enlarged” to a quadratic place Λ: K → L∪∞ such that q has good or fair reduction with respect to Λ in a weak sense, and the splitting properties of q are handed down to Λ_{∗}(q) while there is no form λ_{∗}(q) available for which this would be the case. The details of such a notion of reduction are much more tricky compared to what happens in Chapters I and II. The central term which renders possible a unified theory of generic splitting of quadratic forms is called “stably conservative reduction”, see §31.

One has to get used to the fact that for bilinear forms there is in general no Witt cancellation rule, in contrast to quadratic forms. Nevertheless the specialization theory is in many respects easier for bilinear forms than for quadratic forms.

On the other hand we do not have any theory of generic splitting for symmetric bilinear forms over fields of characteristic 2. Such a theory might not even be possible in a meaningful way. This may well be connected to the fact that the automorphism groups of such forms can be very far from being reductive groups (which may also account for the absence of a good cancellation rule).

This book is intended for audiences with different interests. For a mathematician with perhaps only a little knowledge of quadratic or symmetric bilinear forms, who just wants to get an impression of specialization theory, it suffices to read §1 - §4. The theory of generic splitting in characteristic ≠ 2 will acquaint him with an important application area.

From §5 onwards the book is intended for scholars, working in the algebraic theory of quadratic forms, and also for specialists in the area of algebraic groups. They have always been given something to look at by the theory of quadratic forms.

When a reader has reached §10 of the book, he can lean back in his chair and take a well-deserved break. He has then learned about the specialization theory which is based on the concept of good reduction and has gained a certain perspective on specific phenomena in characteristic 2. Furthermore he has been introduced to the foundations of generic splitting and so has seen the specialization theory in action. Admittedly he has not yet seen independent applications of the weak specialization theory (§3, §7), for this theory has only appeared up to then as an auxiliary one.

The remaining sections §11 - §14 of Chapter II develop the specialization theory sufficiently far to allow an understanding of the classical algebraic theory of quadratic forms (as presented in the books of Lam [L], [L′] and Scharlau [S]) without the usual restriction that the characteristic should be different from 2. Precisely this happens in Chapter III where the reader will also obtain sufficient illustrations, enabling him to relieve other classical theorems from the characteristic ≠ 2 restriction, although this is often a nontrivial task.

The final Chapter IV is ultimately intended for the mathematician who wants to embark on a more daring expedition in the realm of quadratic forms

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over fields. It cannot be a mere coincidence that the specialization theory for quadratic places works just as well as the specialization theory for ordinary places. It is therefore a safe prediction that quadratic places will turn out to be generally useful and important in a future theory of quadratic forms over fields.

Manfred Knebusch Regensburg, June 2007

Postscriptum (October 2009)

I had the very good luck to find a translator of the German text into English, who beside two languages also could understand the mathematical content of this book in depth. I met Professor Thomas Unger within the framework of the European network “Linear Algebraic Groups, Algebraic K-theory, and Related Topics”, and most of the translation and our collaboration has been done under the auspices of this network, which we acknowledge gratefully.

I further owe deep thanks to my former secretary Rosi Bonn, who typed the whole German text in various versions and large parts of the English text.

The German text, Spezialisierung von quadratischen und symmetrischen bilinearen Formen, can be found on my homepage http://www-nw.uni-

regensburg.de/∼.knm22087.mathematik.uni-regensburg.de.

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XII Specialization of Quadratic and Symmetric Bilinear Forms

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Contents

Preface

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII

Chapter I. Fundamentals of Specialization Theory §1 Introduction: on the Problem of Specialization of

Quadratic and Bilinear Forms

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3

§2 An Elementary Treatise on Symmetric Bilinear Forms

. . 7

§3 Specialization of Symmetric Bilinear Forms

. . . . . . . . . . . . . 11

§4 Generic Splitting in Characteristic ≠ 2

. . . . . . . . . . . . . . . . . . 21

§5 An Elementary Treatise on Quadratic Modules

. . . . . . . . . 29

§6 Quadratic Modules over Valuation Rings

. . . . . . . . . . . . . . . . 35

§7 Weak Specialization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

§8 Good Reduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Chapter II. Generic Splitting Theory §9 Generic Splitting of Regular Quadratic Forms

. . . . . . . . . . . 69

§10 Separable Splitting

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

§11 Fair Reduction and Weak Obedience

. . . . . . . . . . . . . . . . . . . . 81

§12 Unified Theory of Generic Splitting

. . . . . . . . . . . . . . . . . . . . . 93

§13 Regular Generic Splitting Towers and Base Extension

. . 99

§14 Generic Splitting Towers of a Specialized Form

. . . . . . . . . 107

Chapter III. Some Applications §15 Subforms which have Bad Reduction

. . . . . . . . . . . . . . . . . . . . 115

§16 Some Forms of Height 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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XIV Contents

§17 The Subform Theorem

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

§18 Milnor’s Exact Sequence

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

§19 A Norm Theorem

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

§20 Strongly Multiplicative Forms

. . . . . . . . . . . . . . . . . . . . . . . . . . 147

§21 Divisibility by Pfister Forms

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

§22 Pfister Neighbours and Excellent Forms

. . . . . . . . . . . . . . . . . 163

§23 Regular Forms of Height 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

§24 Some Open Problems in Characteristic 2

. . . . . . . . . . . . . . . 173

§25 Leading Form and Degree Function

. . . . . . . . . . . . . . . . . . . . . 177

§26 The Companion Form of an Odd-dimensional Regular Form

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

§27 Definability of the Leading Form over the Base Field

… 195

Chapter IV. Specialization with Respect to Quadratic Places

§28 Quadratic Places; Specialization of Bilinear Forms

. . . . . . 201

§29 Almost Good Reduction with Respect to Extensions of Quadratic Places

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

§30 Realization of Quadratic Places; Generic Splitting of Specialized Forms in Characteristic ≠ 2

. . . . . . . . . . . . . . . . . 211

§31 Stably Conservative Reduction of Quadratic Forms

. . . . . 215

§32 Generic Splitting of Stably Conservative Specialized Quadratic Forms

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

References

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Index

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

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Chapter I

Fundamentals of Specialization Theory

2 Chapter I. Fundamentals of Specialization Theory

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§1 Introduction: on the Problem of Specialization of Quadratic and Bilinear Forms

Let φ be a nondegenerate symmetric bilinear form over a field K, in other words

φ(x, y) = ∑_{i,j=1}^{n} a_{ij}x_{i}y_{j},

where x = (x_{1}, …, x_{n}) ∈ K^{n} and y = (y_{1}, …, y_{n}) ∈ K^{n} are vectors, (a_{ij}) is a symmetric (n × n)-matrix with coefficients a_{ij} = a_{ji} ∈ K and det(a_{ij}) ≠ 0. We like to write φ = (a_{ij}). The number of variables n is called the dimension of φ, n = dim φ.

Let also λ: K → L ∪ ∞ be a place, 𝔬 = 𝔬_{λ} the valuation ring associated to K and 𝔪 the maximal ideal of 𝔬. We denote the group of units of 𝔬 by 𝔬^{∗}, 𝔬^{∗} = 𝔬 \ 𝔪.

We would like λ to “specialize” φ to a bilinear form λ_{∗}(φ) over L. When is this possible in a reasonable way? If all a_{ij} ∈ 𝔬 and if det(a_{ij}) ∈ 𝔬^{∗}, then one can associate the nondegenerate form (λ(a_{ij})) over L to φ. This naive idea leads us to the following

Definition. We say that φ has good reduction with respect to λ when φ is isometric to a form (c_{ij}) over K with c_{ij} ∈ 𝔬, det(c_{ij}) ∈ 𝔬^{∗}. We then call the form (λ(c_{ij})) “the” specialization of φ with respect to λ. We denote this specialization by λ_{∗}(φ).

Note. φ = (a_{ij}) is isometric to (c_{ij}) if and only if there exists a matrix S ∈ GL(n, K) with (c_{ij}) = ^{t}S(a_{ij})S. In this case we write φ ≅ (c_{ij}).

We also allow the case dim φ = 0, standing for the unique bilinear form on the zero vector space, the form φ = 0. We agree that the form φ = 0 has good reduction and set λ_{∗}(φ) = 0.

Problem 1. Is this definition meaningful? Up to isometry λ_{∗}(φ) should be independent of the choice of the matrix (c_{ij}).

We will later see that this indeed the case, provided 2 ∉ 𝔪, so that L has characteristic ≠ 2. If L has characteristic 2, then λ_{∗}(φ) is well-defined up to “stable isometry” (see §3).

Problem 2. Is there a meaningful way in which one can associate a symmetric bilinear form over L to φ, when φ has bad reduction?

With regards to this problem we would like to recall a classical result of T.A. Springer, which lets us suspect that finding a solution to the problem is not completely beyond hope. Let v: K → ℤ ∪ ∞ be a discrete valuation of a field K with associated valuation ring 𝔬. Let π be a generator of the maximal ideal 𝔪 of 𝔬, so that 𝔪 = π𝔬. Finally, let k = 𝔬/𝔪 be the residue class field of 𝔬 and λ: K → k ∪ ∞ the canonical place with valuation ring 𝔬. We suppose that 2 ∉ 𝔪, so that char k ≠ 2 is.

Let φ be a nondegenerate symmetric bilinear form over K. Then there exists a decomposition φ ≅ φ_{0} ⊥ πφ_{1}, where φ_{0} and φ_{1} have good reduction with respect to λ. Indeed, we can choose a diagonalisation φ ≅ ⟨a_{1}, …, a_{n}⟩.

\{As usual ⟨a_{1}, …, a_{n}⟩ denotes the diagonal matrix (\begin{matrix}a_{1}  0 \\ . & . & . \\ 0  a_{n} \end{matrix}).\} Then we

can arrange that v(a_{i}) = 0 or 1 for each i, by multiplying the a_{i} by squares and renumbering indices to get a_{i} ∈ 𝔬^{∗} for 1 ≤ i ≤ t and a_{i} = πε_{i}, where ε_{i} ∈ 𝔬^{∗} for t < i ≤ n. {Possibly t = 0, so that φ_{0} = 0, or t = n, so that φ_{1} = 0.}

Theorem (Springer 1955 [Sp]). Let K be complete with respect to the discrete valuation v. If φ is anisotropic (i.e. there is no vector x ≠ 0 in K^{n} with φ(x, x) = 0), then the forms λ_{∗}(φ_{0}) and λ_{∗}(φ_{1}) are anisotropic and up to isometry independent of the choice of decomposition φ ≅ φ_{0} ⊥ πφ_{1}.

Conversely, if ψ_{0} and ψ_{1} are anisotropic forms over k, then there exists up to isometry a unique anisotropic form φ over K with λ_{∗}(φ_{0}) ≅ ψ_{0} and λ_{∗}(φ_{1}) ≅ ψ_{1}.

Given any place λ: K → L∪∞ and any form φ over K, Springer’s theorem suggests to look for a “weak specialization” λ_{W} (φ) by orthogonally decomposing φ in a form φ_{0} with good reduction and a form φ_{1} with “extremely bad” reduction, subsequently forgetting φ_{1} and setting λ_{W} (φ) = λ_{∗}(φ_{0}).

Given an arbitrary valuation ring 𝔬, this sounds like a daring idea. Nonetheless we will see in §3 that a weak specialization can be defined in a meaningful way. Admittedly λ_{W} (φ) is not uniquely determined by φ and λ up to isometry, but up to so-called Witt equivalence. In the situation of Springer’s theorem, λ_{W} (φ) is then the Witt class of φ_{0} and λ_{W} (πφ) the Witt class of φ_{1}.

A quadratic form q of dimension n over K is a function q: K^{n} → K, defined by a homogeneous polynomial of degree 2,

q(x) = ∑_{1≤i≤j≤n} a_{ij}x_{i}x_{j}

(x = (x_{1}, …, x_{n}) ∈ K^{n}). We can associate (a possibly degenerate) symmetric bilinear form

B_{q}(x, y) = q(x + y) − q(x) − q(y) = ∑_{i=1}^{n} 2a_{ii}x_{i}y_{i} + ∑_{i<j} a_{ij}(x_{i}y_{j} + x_{j}y_{i})

to q. It is clear that B_{q}(x, x) = 2q(x) for all x ∈ K^{n}.

§1 Introduction 5

If char K ≠ 2, then any symmetric bilinear form φ over K corresponds to just one quadratic form q over K with B_{q} = φ, namely q(x) = \frac{1}{2} φ(x, x). In this way we can interpret a quadratic form as a symmetric bilinear form and vice versa. In characteristic 2 however, quadratic forms and symmetric bilinear forms are very different objects.

Problem 3. Let λ: K → L ∪ ∞ be a place.

(a) To which quadratic forms q over K can we associate “specialized” quadratic forms λ_{∗}(q) over L in a meaningful way?

(b) Let char L = 2 and char K ≠ 2, hence char K = 0. Should one specialize a quadratic form q over K with respect to λ as a quadratic form, or rather as a symmetric bilinear form?

In what follows we will present a specialization theory for arbitrary nondegenerate symmetric bilinear forms (§3), but only for a rather small class of quadratic forms, the so-called “obedient” quadratic forms (§7). Problem 3(b) will be answered unequivocally. If q is obedient, B_{q} will determine a really boring bilinear form λ_{∗}(B_{q}) (namely a hyperbolic form) which gives almost no information about q. However, λ_{∗}(q) can give important information about q. If possible, a specialization in the quadratic sense is thus to be preferred over a specialization in the bilinear sense.

6 Chapter I. Fundamentals of Specialization Theory

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§2 An Elementary Treatise on Symmetric Bilinear Forms

In this section a “form” will always be understood to be a nondegenerate symmetric bilinear form over a field. So let K be a field.

Theorem 1 (“Witt decomposition”).

(a) Any form φ over K has a decomposition

φ ≅ φ_{0} ⊥ ({a_{1} \atop 1} {1 \atop 0}) ⊥ … ⊥ ({a_{r} \atop 1} {1 \atop 0})

with φ_{0} anisotropic and a_{1}, …, a_{r} ∈ K (r ≥ 0).

(b) The isometry class of φ_{0} is uniquely determined by φ. (Therefore dim φ_{0} and the number r are uniquely determined.)

To clarify these statements, let us recall the following:

(1) A form φ_{0} over K is called anisotropic if φ_{0}(x, x) ≠ 0 for all vectors x ≠ 0.

(2) If char K ≠ 2, then we have for every a ∈ K^{∗} that

({a \atop 1} {1 \atop 0}) ≅ ({0 \atop 1} {1 \atop 0}) ≅ ⟨1, −1⟩ ≅ ⟨a, −a⟩.

Is char K = 2 however and a ≠ 0, then ({a \atop 1} {1 \atop 0}) ≅ ({0 \atop 1} {1 \atop 0}). Indeed if φ = ({0 \atop 1} {1 \atop 0}) we have φ(x, x) = 0 for every vector x ∈ K^{2}, while this is not the case for φ = ({a \atop 1} {1 \atop 0}). In characteristic 2 we still have ({a \atop 1} {1 \atop 0}) ≅ ⟨a, −a⟩ (a ∈ K^{∗}), but ({a \atop 1} {1 \atop 0}) need not be isometric to ({1 \atop 1} {1 \atop 0}) ≅ ⟨1, −1⟩.

(3) The form ({0 \atop 1} {1 \atop 0}) is given the name “hyperbolic plane” (even in characteristic 2), and every form φ, isometric to an orthogonal sum r × ({0 \atop 1} {1 \atop 0}) of r copies of ({0 \atop 1} {1 \atop 0}), is called “hyperbolic” (r ≥ 0).

(4) Forms which are isometric to an orthogonal sum ({a_{1} \atop 1} {1 \atop 0}) ⊥ … ⊥ ({a_{r} \atop 1} {1 \atop 0}) are called metabolic (r ≥ 0). If char K ≠ 2, then every metabolic form is hyperbolic. This is not the case if char K = 2.

(5) If char K = 2, then φ is hyperbolic exactly when every vector x of the underlying vector space K^{n} is isotropic, i.e. φ(x, x) = 0. If φ is not hyperbolic, we can always find an orthogonal basis such that φ ≅ ⟨a_{1}, …, a_{n}⟩ for suitable a_{i} ∈ K^{∗}.

One can find a proof of Theorem 1 in any book about quadratic forms when char K ≠ 2 (see in particular [Bo_{1}], [L], [S]). Part (b) of the theorem is then an immediate consequence of Witt’s Cancellation Theorem. There is no general cancellation theorem in characteristic 2, as the following example shows:

(∗)  ({a \atop 1} {1 \atop 0}) ⊥ ⟨−a⟩ ≅ ({0 \atop 1} {1 \atop 0}) ⊥ ⟨−a⟩

for all a ∈ K^{∗}. If e, f, g is a basis of K^{3} which has the left-hand side of (∗) as value matrix, then e + g, f, g will be a basis which has the right-hand side of (∗) as value matrix. For characteristic 2 one can find proofs of Theorem 1 and the other statements we made in [MH, Chap.I and Chap.III, §1], [K_{1}, §8], [M, §4]. The following is clear from formula (∗):

Lemma 1. If a form φ with dim φ = 2r is metabolic, then there exists a form ψ such that φ ⊥ ψ ≅ r × ({0 \atop 1} {1 \atop 0}) ⊥ ψ.

Definition 1. (a) In the situation of Theorem 1, we call the form φ_{0} the kernel form of φ and r the (Witt) index of φ. We write φ_{0} = ker(φ), r = ind(φ). {In the literature one frequently sees the notation φ_{0} = φ_{an} (“anisotropic part” of φ).}

(b) Two forms φ, ψ over K are called Witt equivalent, denoted by φ ∼ ψ, if ker φ ≅ ker ψ. We write φ ≈ ψ when ker φ ≅ ker ψ and dim φ = dim ψ. On the basis of the next theorem, we then call φ and ψ stably isometric.

Theorem 2. φ ≈ ψ exactly when there exists a form χ such that φ ⊥ χ ≅ ψ ⊥ χ.

We omit the proof. It is easy when one uses Theorem 1, Lemma 1 and the following lemma.

Lemma 2. The form χ ⊥ (−χ) is metabolic for every form χ.

Proof. From Theorem 1(a) we may suppose that χ is anisotropic. If χ is different from the zero form, then χ ≅ ⟨a_{1}, …, a_{n}⟩ with elements a_{i} ∈ K^{∗} (n ≥ 1). Finally, ⟨a_{i}⟩ ⊥ ⟨−a_{i}⟩ ≅ ({a_{i} \atop 1} {1 \atop 0}).

As is well-known, Witt’s Cancellation Theorem (already mentioned above) is valid if char K ≠ 2. It says that two stably isometric forms are already isometric: φ ≈ ψ ⇒ φ ≅ ψ.

Let φ be a form over K. We call the equivalence class of φ with respect to the relation ∼, introduced above, the Witt class of φ and denote it by {φ}. We can add Witt classes together as follows:

{φ} + {ψ}: = {φ ⊥ ψ}.

The class {0} of the zero form, whose members are exactly the metabolic forms, is the neutral element of this addition. From Lemma 2 it follows that

§2 An Elementary Treatise on Symmetric Bilinear Forms 9

{φ} + {−φ} = 0. In this way, the Witt classes of forms over K form an abelian group, which we denote by W (K). We can also multiply Witt classes together:

{φ} · {ψ}: = {φ ⊗ ψ}.

Remark. The definition of the tensor product φ⊗ψ of two forms φ, ψ belongs to the domain of linear algebra [Bo_{1}, §1, No. 9]. For diagonalizable forms we have

⟨a_{1}, …, a_{n}⟩ ⊗ ⟨b_{1}, …, b_{m}⟩ ≅ ⟨a_{1}b_{1}, …, a_{1}b_{m}, a_{2}b_{1}, …, a_{n}b_{m}⟩.

We also have ⟨a_{1}, …, a_{n}⟩ ⊗ ({0 \atop 1} {1 \atop 0}) ≅ n × ({0 \atop 1} {1 \atop 0}). Finally, for a form ({b \atop 1} {1 \atop 0}) with b ≠ 0 we have

⟨a⟩ ⊗ ({b \atop 1} {1 \atop 0}) ≅ ⟨a⟩ ⊗ ⟨b, −b⟩ ≅ ⟨ab, −ab⟩ ≅ ({ab \atop 1} {1 \atop 0}) .

Now it is clear that the tensor product of any given form and a metabolic form is again metabolic. {For a conceptual proof of this see [K_{1}, §3], [MH, Chap. I].} Therefore the Witt class {φ ⊗ ψ} is completely determined by the classes {φ}, {ψ}, independent of the choice of representatives φ, ψ.

With this multiplication, W (K) becomes a commutative ring. The identity element is {⟨1⟩}. We call W (K) the Witt ring of K. For char K ≠ 2 this ring was already introduced by Ernst Witt in 1937 [W].

We would like to describe the ring W (K) by generators and relations. In characteristic ≠ 2 this was already known by Witt [oral communication] and is implicitly contained in his work [W, Satz 7].

First we must recall the notion of determinant of a form. For a ∈ K^{∗}, the isometry class of a one-dimensional form ⟨a⟩ will again be denoted by ⟨a⟩. The tensor product ⟨a⟩ ⊗ ⟨b⟩ will be abbreviated by ⟨a⟩⟨b⟩. We have ⟨a⟩⟨b⟩ = ⟨ab⟩ and ⟨a⟩⟨a⟩ = ⟨1⟩. In this way the isometry classes form an abelian group of exponent 2, which we denote by Q(K). Given a, b ∈ K^{∗}, it is clear that ⟨a⟩ = ⟨b⟩ exactly when b = ac^{2} for a c ∈ K^{∗}. So Q(K) is just the group of square classes K^{∗}/K^{∗2} in disguise. We identify Q(K) = K^{∗}/K^{∗2}.

It is well-known that for a given form φ = (a_{ij}) the square class of the determinant of the symmetric matrix (a_{ij}) only depends on the isometry class of φ. We denote this square class by det(φ), so det(φ) = ⟨det(a_{ij})⟩, and call it the determinant of φ. A slight complication arises from the fact that the determinant is not compatible with Witt equivalence. To remedy this, we introduce the signed determinant

d(φ): = ⟨−1⟩^{\frac{n(n−1)}{2}} · det(φ)

(n: = dim φ). One can easily check that d(φ ⊥ ({a \atop 1} {1 \atop 0})) = d(φ), for any a ∈ K. Hence d(φ) depends only on the Witt class {φ}. The signed determinant d(φ) also has a disadvantage though. In contrast with det(φ), d(φ) does not behave completely well with respect to the orthogonal sum. Let ν(φ) denote the dimension index of φ, ν(φ) = dim φ + 2ℤ ∈ ℤ/2ℤ. Then we have (cf. [S, I §2])

d(φ ⊥ ψ) = ⟨−1⟩^{ν(φ)ν(ψ)}d(φ)d(ψ).

Let us now describe W (K) by means of generators and relations. Every one-dimensional form ⟨a⟩ satisfies d(⟨a⟩) = ⟨a⟩. This innocent remark shows that the map from Q(K) to W (K), which sends every isometry class ⟨a⟩ to its Witt class {⟨a⟩}, is injective. We can thus interpret Q(K) as a subgroup of the group of units of the ring W (K), Q(K) ⊂ W (K)^{∗}.

W (K) is additively generated by the subset Q(K), since every nonhyperbolic form can be written as ⟨a_{1}, …, a_{n}⟩ = ⟨a_{1}⟩ ⊥ … ⊥ ⟨a_{n}⟩. Hence Q(K) is a system of generators of W (K). There is an obviously surjective ring homomorphism

Φ: ℤ[Q(K)] ↠ W (K)

from the group ring ℤ[Q(K)] to W (K). Recall that ℤ[Q(K)] is the ring of formal sums ∑_{g} n_{g}g with g ∈ Q(K), n_{g} ∈ ℤ, and almost all n_{g} = 0. Φ

associates to such a sum the in W (K) constructed sum ∑_{g} n_{g}g.

The elements of the kernel of Φ are the relations on Q(K) we are looking for. We can write down some of those relations immediately: for every a ∈ K^{∗} is ⟨a⟩ + ⟨−a⟩ clearly a relation. For a, b ∈ K^{∗} and given λ, μ ∈ K^{∗}, the form ⟨a, b⟩ represents the element c: = λ^{2}a + μ^{2}b. If c ≠ 0, then we can find another element d ∈ K^{∗} with ⟨a, b⟩ ≅ ⟨c, d⟩ Comparing determinants shows that ⟨d⟩ = ⟨abc⟩. Hence ⟨a⟩ + ⟨b⟩ − ⟨c⟩ − ⟨abc⟩ = (⟨a⟩ + ⟨b⟩)(⟨1⟩ − ⟨c⟩) is also a relation. We have the technically important

Theorem 3. The ideal Ker Φ of the ring ℤ[Q(K)] is additively generated (i.e. as abelian group) by the elements ⟨a⟩ + ⟨−a⟩, a ∈ K^{∗} and the elements ⟨a⟩+⟨b⟩−⟨c⟩−⟨abc⟩ with a, b ∈ K^{∗}, ⟨b⟩ ≠ ⟨−a⟩, c = λ^{2}a+μ^{2}b with λ, μ ∈ K^{∗}.

Remark. Ker Φ is therefore generated as an ideal by the element ⟨1⟩ + ⟨−1⟩ and the elements (⟨1⟩+⟨a⟩)(1−⟨c⟩) with ⟨a⟩ ≠ ⟨−1⟩, c = 1+λ^{2}a with λ ∈ K^{∗}. For application in the next section, the additive description of Ker Φ above is more favourable though.

A proof of Theorem 3, which also works in characteristic 2, can be found in [K_{1}, §5], [KRW, §1], [K_{2} II, §4] (even over semi-local rings instead of over fields^{2}), [MH, p.85]. For characteristic ≠ 2 the proof is a bit simpler, since every form has an orthogonal basis in this case, see [S, I § 9].

2 The case where K has only 2 elements, K = 𝔽_{2}, is not covered by the more general theorems there. The statement of Theorem 3 for K = 𝔽_{2} is trivial however, since K has only one square class ⟨1⟩ and ⟨1, 1⟩ ∼ 0.

§3 Specialization of Symmetric Bilinear Forms

In this section, a “form” will again be understood to be a nondegenerate symmetric bilinear form. Let λ: K → L ∪ ∞ be a place from the field K to a field L. Let 𝔬 = 𝔬_{λ} be the valuation ring associated to λ and 𝔪 its maximal ideal. As usual for rings, 𝔬^{∗} stands for the group of units of 𝔬, so that 𝔬^{∗} = 𝔬 \ 𝔪. This is the set of all x ∈ K with λ(x) ≠ 0, ∞.

We will now denote the Witt class of a one-dimensional form ⟨a⟩ over K (or L) by {a}. The group of square classes Q(𝔬) = 𝔬^{∗}/𝔬^{∗2} can be embedded in Q(K) = K^{∗}/K^{∗2} in a natural way via a𝔬^{∗2} ↦ aK^{∗2}. We interpret Q(𝔬) as a subgroup of Q(K), so Q(𝔬) = {⟨a⟩ | a ∈ 𝔬^{∗}} ⊂ Q(K). Our specialization theory is based on the following

Theorem 1. There exists a well-defined additive map λ_{W}: W (K) → W (L), given by λ_{W} ({a}) = {λ(a)} if a ∈ 𝔬^{∗}, and λ_{W} ({a}) = 0 if ⟨a⟩ ∉ Q(𝔬) (i.e. (aK^{∗2}) ∩ 𝔬^{∗} = ∅).^{3}

Proof. (Copied from [K_{4}, §3].) Our place λ is a combination of the canonical place K → (𝔬/𝔪) ∪ ∞ with respect to 𝔬, and a field extension \overline{λ}: 𝔬/𝔪 ↪ L. Thus it suffices to prove the theorem for the canonical place. So let L = 𝔬/𝔪 and λ(a) = \overline{a}: = a + 𝔪 for a ∈ 𝔬.

We have a well-defined additive map Λ: ℤ[Q(K)] → W (L) such that Λ(⟨a⟩) = \{\overline{a}\} if a ∈ 𝔬^{∗}, and Λ(⟨a⟩) = 0 if ⟨a⟩ ∉ Q(𝔬). Clearly Λ vanishes on all elements ⟨a⟩ + ⟨−a⟩ with a ∈ K^{∗}. According to §2, Theorem 3 we will be finished if we can show that Λ also disappears on every element

Z = ⟨a_{1}⟩ + ⟨a_{2}⟩ − ⟨a_{3}⟩ − ⟨a_{4}⟩

with a_{i} ∈ K^{∗} and ⟨a_{1}, a_{2}⟩ ≅ ⟨a_{3}, a_{4}⟩.

This will be the case when the four square classes ⟨a_{i}⟩ are not all in Q(𝔬). Suppose from now on, without loss of generality, that a_{1} ∈ 𝔬^{∗}. Then we have Z = ⟨a_{1}⟩y, where

y = 1 + ⟨c⟩ − ⟨b⟩ − ⟨bc⟩

is an element such that ⟨1, c⟩ ≅ ⟨b, bc⟩. So b = u^{2} +w^{2}c for elements u, w ∈ K. Clearly the equation Λ(⟨a⟩x) = \{\overline{a}\}Λ(x) is satisfied for any a ∈ 𝔬^{∗}, x ∈ ℤ[Q(K)]. Therefore it is enough to verify that Λ(y) = 0. We suppose without loss of generality that u and w are not both zero, otherwise we already have that y = 0.

3 The letter W in the notation λ_{W} refers to “Witt” or “weak”, see §1 and §7.

Let us first treat the case ⟨c⟩ ∈ Q(𝔬), so without loss of generality c ∈ 𝔬^{∗}. Then we have

Λ(y) = (1 + \{\overline{c}\})Λ(1 − ⟨b⟩).

If \{\overline{c}\} = \{−1\}, we are done. So suppose from now on that \{\overline{c}\} ≠ \{−1\}. Then the form ⟨1, \overline{c}⟩ is anisotropic over L. Since we are allowed to replace u and v by gu and gv for some g ∈ K^{∗}, we may additionally assume that u and v are both in 𝔬, but not both in 𝔪. Since ⟨1, \overline{c}⟩ is anisotropic, we have \overline{b} = \overline{u}^{2} + \overline{c}\overline{w}^{2} ≠ 0 and

Λ(y) = (1 + \{\overline{c}\})(1 − \{\overline{u}^{2} + \overline{c}\overline{w}^{2}\}) = 0.

The case which remains to be tackled is when the square class cK^{∗2} doesn’t contain a unit from 𝔬. Then u^{−2}w^{2}c is definitely not a unit and either b = u^{2}(1 + d) or b = w^{2}c(1 + d) with d ∈ 𝔪. Hence Λ(1 − ⟨\overline{b}⟩) is 0 or 1 − \{\overline{c}\}, and both times Λ(y) = 0.

Scholium. The map λ_{W}: W (K) → W (L) can be described very conveniently as follows: Let φ be a form over K. If φ is hyperbolic (or, more generally metabolic), then λ_{W} ({φ}) = 0. If φ is not hyperbolic, then consider a diagonalisation φ ≅ ⟨a_{1}, a_{2}, …, a_{n}⟩. Multiply each coefficient a_{i}, for which it is possible, by a square so that it becomes a unit in 𝔬, and leave the other coefficients as they are. Let for example a_{i} ∈ 𝔬^{∗} for 1 ≤ i ≤ r and ⟨a_{i}⟩ ∉ Q(𝔬) for r < i ≤ n (possibly r = 0 or r = n). Then λ_{W} ({φ}) = {⟨λ(a_{1}), …, λ(a_{r})⟩}.

Let us now recall a definition from the Introduction §1.

Definition 1. We say that a form φ over K has good reduction with respect to λ, or that φ is λ-unimodular if φ is isometric to a form (a_{ij}) with a_{ij} ∈ 𝔬 and det(a_{ij}) ∈ 𝔬^{∗}. We call such a representation φ ≅ (a_{ij}) a λ-unimodular represention of φ (or a unimodular representation with respect to the valuation ring 𝔬).

This definition can be interpreted geometrically as follows. We associate to φ a couple (E, B), consisting of an n-dimensional K-vector space E (n = dim φ) and a symmetric bilinear form B: E × E → K such that B represents the form φ after a choice of basis of E. We denote this by φ ˆ= (E, B). Since φ has good reduction with respect to λ, E contains a free 𝔬-submodule M of rank n with E = KM, i.e. E = K ⊗_{𝔬} M, and with B(M × M) ⊂ 𝔬, such that the restriction B|M × M: M × M → 𝔬 is a nondegenerate bilinear form over 𝔬, i.e. gives rise to an isomorphism x ↦ B(x, −) from the 𝔬-module M

to the dual 𝔬-module M^{∨} = Hom_{𝔬}(M, 𝔬).

By means of Theorem 1 we can now quite easily find a solution of the first problem posed in §1.

Theorem 2. Suppose that the form φ over K has good reduction with respect to λ. Let φ ≅ (a_{ij}) be a unimodular representation of φ. Then the Witt class λ_{W} ({φ}) is represented by the (nondegenerate! ) form (λ(a_{ij})) over L.

§3 Specialization of Symmetric Bilinear Forms 13

Consequently the form (λ(a_{ij})) is up to stable isometry independent of the choice of unimodular representation. (Recall that, if two forms ψ and ψ′ are Witt equivalent and dim ψ = dim ψ′, then ψ ≈ ψ′.)

To prove this theorem, we need the following easy lemma about lifting orthogonal bases.

Lemma. Let M be a finitely generated free 𝔬-module, equipped with a nondegenerate symmetric bilinear form B: M × M → 𝔬. Let k: = 𝔬/𝔪 and let π: M → M/𝔪M be the natural epimorphism from M to the k-vector space M/𝔪M. Further, let \overline{B} be the (again nondegenerate) bilinear form induced by B on M/𝔪M, \overline{B}(π(x), π(y)): = B(x, y) + 𝔪. Suppose that the vector space M/𝔪M has a basis \overline{e}_{1}, …, \overline{e}_{n}, orthogonal with respect to \overline{B}. Then M has a basis e_{1}, …, e_{n}, orthogonal with respect to B, with π(e_{i}) = \overline{e}_{i} (1 ≤ i ≤ n).

Proof. By induction on n, which obviously is the rank of the free 𝔬-module M. For n = 1 nothing has to be shown. So suppose that n > 1. We choose an element e_{1} ∈ M with π(e_{1}) = \overline{e}_{1}. Then B(e_{1}, e_{1}) ∈ 𝔬^{∗} since \overline{B}(\overline{e}_{1}, \overline{e}_{1}) ≠ 0. Hence the restriction of B to the module 𝔬e_{1} is a nondegenerate bilinear form on 𝔬e_{1}. Invoking a very simple theorem (e.g. [MH, p.5, Th.3.2], §5, Lemma 1 below) yields M = (𝔬e_{1}) ⊥ N with N = (𝔬e_{1})^{⊥} = {x ∈ M | B(x, e_{1}) = 0}. The restriction π|N: N → M/𝔪M is then a homomorphism from N to

(k\overline{e}_{1})^{⊥} = ⨁_{i=2}^{n} k\overline{e}_{i} with kernel 𝔪N. By our induction hypothesis, N contains

an orthogonal basis e_{2}, …, e_{n} with π(e_{i}) = \overline{e}_{i} (2 ≤ i ≤ n) which can be completed by e_{1} to form an orthogonal basis of M which has the required property.

Remark. Clearly the lemma and its proof remain valid when 𝔬 is an arbitrary local ring with maximal ideal 𝔪, instead of a valuation ring.

We also need the following

Definition 2. A bilinear 𝔬-module is a couple (M, B) consisting of an 𝔬module M and a symmetric bilinear form B: M ×M → 𝔬. A bilinear module is called free when the 𝔬-module M is free of finite rank. If e_{1}, …, e_{n} is a basis of M, we write (M, B) ≅ (a_{ij}) with a_{ij}: = B(e_{i}, e_{j}). If e_{1}, …, e_{n} is an orthogonal basis (B(e_{i}, e_{j}) = 0 for i ≠ j), then we also write (M, B) ≅ ⟨a_{1}, …, a_{n}⟩ with a_{i}: = B(e_{i}, e_{i}).

Note. The form B is nondegenerate exactly when det(a_{ij}) is a unit in 𝔬, respectively when all a_{i} are units in 𝔬.

All this makes sense and remains correct when 𝔬 is an arbitrary commutative ring (with 1), instead of a valuation ring. As before, “≅” stands for “isometric”, also for bilinear modules.

Proof of Theorem 2. For a ∈ 𝔬, let \overline{a} denote the image of a in 𝔬/𝔪. We suppose for the moment that the bilinear space (\overline{a}_{ij}) over 𝔬/𝔪 is not hyperbolic. Then it has an orthogonal basis. By the lemma, the bilinear module (a_{ij}) over 𝔬 also has an orthogonal basis. Hence over 𝔬,

(∗)  (a_{ij}) ≅ ⟨a_{1}, …, a_{n}⟩

for certain a_{i} ∈ 𝔬^{∗}. The isometry (∗) is then also valid over K, and so we have in W (L)

λ_{W} ({φ}) = {⟨λ(a_{1}), …, λ(a_{n})⟩}.

On the other hand (∗) implies that

(\overline{a}_{ij}) ≅ ⟨\overline{a}_{1}, …, \overline{a}_{n}⟩ over 𝔬/𝔪.

If we now apply the (injective) homomorphism \overline{λ}: 𝔬/𝔪 → L induced by λ (thus we tensor with the field extension given by \overline{λ}), we obtain

(λ(a_{ij})) ≅ ⟨λ(a_{1}), …, λ(a_{n})⟩

over L. Consequently the Witt class λ_{W} ({φ}) is represented by the form (λ(a_{ij})).

Let us now tackle the remaining case, where the form (\overline{a}_{ij}) over 𝔬/𝔪 is hyperbolic. We can apply what we just have proved to the form ψ: = φ ⊥ ⟨1⟩. This gives us

\begin{aligned}λ_{W} (\{ψ\}) & = \{(λ(a_{ij})) ⊥ ⟨1⟩\} \\ & = \{(λ(a_{ij}))\} + \{⟨1⟩\} \end{aligned}

in W (L). On the other hand we have λ_{W} ({ψ}) = λ_{W} ({φ}) + {⟨1⟩}, and we find again that λ_{W} ({φ}) = {(λ(a_{ij}))}.

Remark. If char L ≠ 2, a hyperbolic form over 𝔬/𝔪 also has an orthogonal basis, so that the distinction between the two cases above is unnecessary.

Definition 3. If φ has good reduction with respect to λ, φ ≅ (a_{ij}) with a_{ij} ∈ 𝔬, det(a_{ij}) ∈ 𝔬^{∗}, we denote the form (λ(a_{ij})) over L by λ_{∗}(φ) and call it “the” specialization of φ with respect to λ.

If char L = 2 we run into trouble with this definition, since λ_{∗}(φ) is only up to stable isometry uniquely determined by φ . We nevertheless use it, since it is so convenient. If char L ≠ 2, λ_{∗}(φ) is up to isometry uniquely determined by φ.

Example. Every metabolic form φ over K has good reduction with respect to λ. Of course is λ_{∗}(φ) ∼ 0.

Proof. It suffices to prove this in the case dim φ = 2, so φ = ({a \atop 1} {1 \atop 0}) with a ∈ K. Let φˆ=(E, B) and let e, f be a basis of E with value matrix ({a \atop 1} {1 \atop 0}). Choose an element c ∈ K^{∗} with ac^{2} ∈ 𝔬. Then ce, c^{−1}f is a basis of E with value matrix ({ac^{2} \atop 1} {1 \atop 0}).

Theorem 3. Let φ and ψ be forms over K, having good reduction with respect to λ. Then φ ⊥ ψ also has good reduction with respect to λ and

λ_{∗}(φ ⊥ ψ) ≈ λ_{∗}(φ) ⊥ λ_{∗}(ψ).

Proof. This is clear.

Until now we got on with our specialization theory almost without any knowledge of bilinear forms over 𝔬. Except for the lemma above about the existence of orthogonal bases, we needed hardly anything from this area. We could even have avoided using this little bit of information if we would only have considered diagonalised forms over fields.

We are still missing one important theorem of specialization theory (especially for applications later on), Theorem 4 below. For a proof of this theorem we need the basics of the theory of forms over valuation rings, which we will present next using a “geometric” point of view. In other words, we interpret a form φ over a field as an “inner product” on a vector space and use more generally “inner products” on modules over rings, while until now a form was usually interpreted as a polynomial in two sets of variables x_{1}, …, x_{n}, y_{1}, …, y_{n}.

For the moment we allow local rings instead of the valuation ring 𝔬 since this won’t cost us anything extra. So let A be a local ring.

Definition 4. A bilinear space M over A is a free A-module M of finite rank, equipped with a symmetric bilinear form B: M × M → A which is nondegenerate, i.e. which determines an isomorphism x ↦ B(x, −) from M

on the dual module M^{∨} = Hom_{A}(M, A).

Remark. We usually denote a bilinear space by the letter M. If confusion is possible, we write (M, B) or even (M, B_{M}).

In what follows, M denotes a bilinear space over A, with associated bilinear form B.

Definition 5. A subspace V of M is a submodule V of M which is a direct summand of M, i.e. for which there exists another submodule W of M with M = V ⊕ W.

To a subspace V we can associate the orthogonal submodule

V^{⊥} = {x ∈ M | B(x, V) = 0},

and we have an exact sequence

0 ⟶ V^{⊥} ⟶ M ⟶^{φ} V^{∨} ⟶ 0.

Here V^{∨} = Hom_{A}(V, A) and φ maps x ∈ M to the linear form y ↦ B(x, y)

on V. The sequence splits since V^{∨} is free. Thus V^{⊥} is again a subspace of M.

Definition 6. A subspace V of M is called totally isotropic when B(V, V) = {0}, i.e. when V ⊂ V^{⊥}. V is called a Lagrangian subspace of M when V = V^{⊥}. If M contains a Lagrangian subspace, M is called metabolic. M is called anisotropic if it doesn’t contain any totally isotropic subspace V ≠ {0}.

Lemma 1.

(a) Every bilinear space M over A has a decomposition

M ≅ M_{0} ⊥ M_{1}

with M_{0} anisotropic and M_{1} metabolic.

(b) Every metabolic space N over A is the orthogonal sum of spaces of the form ({a \atop 1} {1 \atop 0}),

N ≅ ({a_{1} \atop 1} {1 \atop 0}) ⊥ … ⊥ ({a_{r} \atop 1} {1 \atop 0})

with a_{1}, …, a_{r} ∈ A.

These statements can be inferred from more general theorems, which can be found in e.g. [Ba, §1], [K_{1}, §3], [K_{2}, I §3], [KRW, §1].

Remark. If 2 is a unit in A, Witt’s Cancellation Theorem ([K_{3}], [R]) holds for bilinear spaces over A and every metabolic space over A is even hyperbolic, i.e. is an orthogonal sum r × ({0 \atop 1} {1 \atop 0}) of copies of the “hyperbolic plane” ({0 \atop 1} {1 \atop 0}) over A. Now the anisotropic space M_{0} in Lemma 1(a) is up to isometry uniquely determined by M. If 2 ∉ A^{∗} this is false in general.

If α: A → C is a homomorphism from A to another local ring C, we can associate to a bilinear space (M, B) = M over A a bilinear space (C ⊗_{A} M, B′) = C ⊗_{A} M over C as follows: the underlying free C-module is the tensor product C ⊗_{A} M determined by α, and the C-bilinear form B′ on this module is obtained from B by means of a basis extension, so

B′(c ⊗ x, d ⊗ y) = cd α(B(x, y))

(x, y ∈ M; c, d ∈ C). The form B′ is again nondegenerate. If (a_{ij}) is the value matrix of B with respect to a basis e_{1}, …, e_{n} of M, then (α(a_{ij})) is the value matrix of B′ with respect to the basis 1 ⊗ e_{1}, …, 1 ⊗ e_{n} of C ⊗_{A} M.

If A doesn’t contain any zero divisors and if K is the quotient field of A, we can in particular use the inclusion A ↪ K to associate a bilinear space K ⊗_{A} M to the bilinear space M over A. Now we can interpret M as an A-submodule of the K-vector space K ⊗_{A} M (x = 1 ⊗ x for x ∈ M) and reconstruct B from B′ by restriction, B = B′|M × M: M × M → A.

Let us return to our place λ: K → L ∪ ∞ and the valuation ring 𝔬.

Lemma 2. Let M be a bilinear space over 𝔬.

(a) If K ⊗_{𝔬} M is isotropic, then M is isotropic.

(b) If K ⊗_{𝔬} M is metabolic, then M is metabolic.

Proof. Let E: = K ⊗_{𝔬} M. We interpret M as an 𝔬-submodule of E and have E = KM.

(a) If E is isotropic, there exists a subspace W ≠ {0} in E with W ⊂ W^{⊥}. The 𝔬-submodule V: = W ∩ M of M satisfies KV = W and so V ≠ {0}. Furthermore V ⊂ V^{⊥}. The 𝔬-module M/V is torsion free and finitely generated, hence free. This is because every finitely generated ideal in 𝔬 is principal, cf. [CE, VII, §4]. Therefore V is a totally isotropic subspace of E.

(b) If W = W^{⊥}, then V = V^{⊥}. Hence M is metabolic.

Now we are fully equipped to prove the following important theorem [K_{4}, Prop.2.2].

Theorem 4. Let φ and ψ be forms over K. If φ and φ ⊥ ψ have good reduction with respect to λ, then ψ also has good reduction with respect to λ.

Proof. Adopting geometric language, the statement says: Let F and G be bilinear spaces over K and E: = F ⊥ G. If F and E have good reduction, i.e. F ≅ K ⊗_{𝔬} N, E ≅ K ⊗_{𝔬} M for bilinear spaces N and M over 𝔬, then G has good reduction as well.

By §2, Theorem 1 there is a decomposition G = G_{0} ⊥ G_{1} with G_{0} anisotropic and G_{1} metabolic. From above (cf. the example following Definition 3), G_{1} has good reduction. Hence it suffices to show that G_{0} has good reduction.

Now E ⊥ (−F) ≅ F ⊥ (−F) ⊥ G_{0} ⊥ G_{1}.^{4} Since F ⊥ (−F) ⊥ G_{0} is metabolic, but G_{1} anisotropic, G_{1} is the kernel space of E ⊥ (−F). (“Kernel space” is the pendant of the word “kernel form” (= anisotropic part) in the geometric language.) We decompose M ⊥ (−N) following Lemma 1(a), M ⊥ (−N) ≅ R ⊥ S where R is anisotropic and S metabolic. Tensoring with K gives E ⊥ (−F) ≅ K ⊗_{𝔬} R ⊥ K ⊗_{𝔬} S. Now K ⊗_{𝔬} S is metabolic and, according to Lemma 2, K ⊗_{𝔬} R is anisotropic. Hence K ⊗_{𝔬} R is also a kernel space of E ⊥ (−F). Applying §2, Theorem 1 gives K ⊗_{𝔬} R ≅ G_{0}, and we are finished.

Corollary. Let φ and ψ be forms over K with φ ∼ ψ. If φ has good reduction with respect to λ, ψ also has good reduction with respect to λ and λ_{∗}(φ) ∼ λ_{∗}(ψ). If furthermore φ ≈ ψ, then λ_{∗}(φ) ≈ λ_{∗}(ψ).

Proof. There are Witt decompositions φ ≅ φ_{0} ⊥ μ, ψ ≅ ψ_{0} ⊥ ν with φ_{0}, ψ_{0} anisotropic and μ, ν metabolic. As established above, μ and ν have good reduction with respect to λ and λ_{∗}(μ), λ_{∗}(ν) are metabolic. By assumption φ has good reduction with respect to λ and φ_{0} is isometric to ψ_{0}. Theorem 4 implies that φ_{0} has good reduction with respect to λ. Therefore ψ_{0}, and hence ψ, has good reduction with respect to λ, and (according to Theorem 3)

\begin{aligned}λ_{∗}(φ) & ≈ λ_{∗}(φ_{0}) ⊥ λ_{∗}(μ) ∼ λ_{∗}(φ_{0}), \\ λ_{∗}(ψ) & ≈ λ_{∗}(ψ_{0}) ⊥ λ_{∗}(ν) ∼ λ_{∗}(ψ_{0}). \end{aligned}

4 If E = (E, B) is a bilinear space, then −E denotes the space (E, −B).

Naturally λ_{∗}(φ_{0}) ≈ λ_{∗}(ψ_{0}), so λ_{∗}(φ) ∼ λ_{∗}(ψ). If φ ≈ ψ, then φ and ψ have the same dimension and so λ_{∗}(φ) ≈ λ_{∗}(ψ).

Let us give a small illustration of Theorem 4.

Definition 7. Let φ and ψ be forms over a field K. If there exists a form χ over K with φ ≅ ψ ⊥ χ, we say that φ represents the form ψ and write ψ < φ.

For example, the one-dimensional forms represented by φ are exactly the square classes ⟨φ(x, x)⟩, where x runs through the anisotropic vectors of the space belonging to φ.

Theorem 5 (Substitution Principle). Let k be a field and K = k(t), where t = (t_{1}, …, t_{r}) is a set of indeterminates. Let (f_{ij}(t)) be a symmetric (n×n)- matrix and (g_{kl}(t)) a symmetric (m×m)-matrix, for polynomials f_{ij}(t) ∈ k[t], and g_{kl}(t) ∈ k[t]. Let further be given a field extension k ⊂ L and a point c ∈ L^{r} with det(f_{ij}(c)) ≠ 0 and det(g_{kl}(c)) ≠ 0. If char k = 2, also suppose that the form (f_{ij}(c)) is anisotropic over L.

Claim: if (g_{kl}(t)) < (f_{ij}(t)) (as forms over K), then (g_{kl}(c)) < (f_{ij}(c)) (as forms over L).

Proof. Going from k[t] to L[t], we suppose without loss of generality that L = k. For every s ∈ {1, …, r} there is exactly one corresponding place λ_{s}: k(t_{1}, …, t_{s}) → k(t_{1}, …, t_{s−1}) ∪ ∞ with λ_{s}(u) = u for all u ∈ k(t_{1}, …, t_{s−1}) and λ_{s}(t_{s}) = c_{s}. {Read k(t_{1}, …, t_{s−1}) = k when s = 1.} The composition λ_{1} ∘ λ_{2} ∘ ⋯ ∘ λ_{s} of these places is a place λ: K → k ∪ ∞ with λ(a) = a for all a ∈ k and λ(t_{i}) = c_{i} for i = 1, …, r. Let φ denote the form (f_{ij}(t)) over K and ψ the form (g_{kl}(t)) over K. {Note that obviously det(f_{ij}(t)) ≠ 0, det(g_{kl}(t)) ≠ 0.} The forms φ and ψ both have good reduction with respect to λ and λ_{∗}(φ) ≈ (f_{ij}(c)), λ_{∗}(ψ) ≈ (g_{kl}(c)).

Now let ψ < φ. Then there exists a form χ over K with ψ ⊥ χ ≅ φ. According to Theorem 4, χ has good reduction with respect to λ and according to Theorem 3, λ_{∗}(ψ) ⊥ λ_{∗}(χ) ≈ λ_{∗}(φ). Hence λ_{∗}(ψ) ⊥ λ_{∗}(χ) ≅ λ_{∗}(φ) if char k ≠ 2. If char k = 2 and λ_{∗}(φ) is anisotropic, this remains true, since λ_{∗}(φ) is up to isometry the unique anisotropic form in the Witt class λ_{W} (φ), and λ_{∗} (ψ) ⊥ λ_{∗} (χ) has the same dimension as λ_{∗}(φ).

Let us now return to our arbitrary place λ: K → L ∪ ∞ and to the conventions made at the beginning of the paragraph. The Lemmas 1 and 2 allow us to give an easier proof of Theorem 2, which is interesting in its own right.

Second proof of Theorem 2. We adopt the geometric language. Let E be a bilinear space over K, having good reduction with respect to λ and let M and N be bilinear spaces over 𝔬 with E ≅ K ⊗_{𝔬} M ≅ K ⊗_{𝔬} N. We have to show that L ⊗_{λ} M ≈ L ⊗_{λ} N, where the tensor product is taken over 𝔬, and L is regarded as an 𝔬-algebra via the homomorphism λ|𝔬 : 𝔬 → L. The space K ⊗_{𝔬} (M ⊥ (−N)) is metabolic. According to Lemma 2, M ⊥ (−N) is metabolic. Hence L ⊗_{λ} (M ⊥ (−N)) = L ⊗_{λ} M ⊥ (−L ⊗_{λ} N) is also metabolic. Therefore L ⊗_{λ} M ≈ L ⊗_{λ} N.

We can now describe the property “good reduction” and the specialization of a form by means of diagonal forms as follows.

Theorem 6. Let φ be a form over K, dim φ = n.

(a) The form φ has good reduction with respect to λ if and only if φ is Witt equivalent to a diagonal form ⟨a_{1}, …, a_{r}⟩ with units a_{i} ∈ 𝔬^{∗}. In this case dim λ_{∗}(φ) = n and λ_{∗}(φ) ∼ ⟨λ(a_{1}), …, λ(a_{r})⟩. Furthermore one can choose r = n + 2.

(b) Let 2 ∈ 𝔬^{∗}, i.e. char L ≠ 2. The form φ has good reduction with respect to λ if and only if φ is isometric to a diagonal form ⟨a_{1}, …, a_{n}⟩ with a_{i} ∈ 𝔬^{∗}. In this case λ_{∗}(φ) ≅ ⟨λ(a_{1}), …, λ(a_{n})⟩.

Proof of part (a). If φ ∼ ⟨a_{1}, …, a_{r}⟩ with units a_{i} ∈ 𝔬^{∗}, then by the Corollary φ has good reduction with respect to λ and λ_{∗}(φ) ∼ ⟨λ(a_{1}), …, λ(a_{r})⟩. Suppose now that φ has good reduction with respect to λ. The form φ corresponds to a bilinear space K ⊗_{𝔬} M over K, which comes from a bilinear space M over 𝔬. If M/𝔪M is not hyperbolic, then M has an orthogonal basis by the Lemma following Theorem 2. Therefore φ ≅ ⟨a_{1}, …, a_{n}⟩ with units a_{i} ∈ 𝔬^{∗}. In general we consider the space M′: = M ⊥ ⟨1, −1⟩ over 𝔬. Then M′/𝔪M′ is definitely not hyperbolic. Hence φ ⊥ ⟨1, −1⟩ ≅ ⟨b_{1}, …, b_{n+2}⟩ with units b_{i}.

The proof of part (b) is similar, but simpler since now M/𝔪M always has an orthogonal basis. We don’t need the Corollary here.

Finally, we consider the specialization of tensor products of forms.

Theorem 7. Let φ and ψ be two forms over K, which have good reduction with respect to λ. Then φ ⊗ ψ also has good reduction with respect to λ, and λ_{∗}(φ ⊗ ψ) ≈ λ_{∗}(φ) ⊗ λ_{∗}(ψ).

Proof. According to Theorem 6 we have the following Witt equivalences, φ ∼ ⟨a_{1}, …, a_{m}⟩, ψ ∼ ⟨b_{1}, …, b_{n}⟩, with units a_{i}, b_{j} ∈ 𝔬^{∗}. Then φ ⊗ ψ ∼ ⟨a_{1}b_{1}, a_{1}b_{2}, …, a_{1}b_{n}, …, a_{m}b_{n}⟩. Again according to Theorem 6, φ ⊗ ψ has good reduction and

λ_{∗}(φ ⊗ ψ) ∼ ⟨λ(a_{1})λ(b_{1}), …, λ(a_{m})λ(b_{n})⟩
≅ ⟨λ(a_{1}), …, λ(a_{m})⟩ ⊗ ⟨λ(b_{1}), …, λ(b_{n})⟩
∼ λ_{∗}(φ) ⊗ λ_{∗}(ψ).

Now the forms λ_{∗}(φ ⊗ ψ) and λ_{∗}(φ) ⊗ λ_{∗}(ψ) both have the same dimension as φ ⊗ ψ. Therefore λ_{∗}(φ ⊗ ψ) ≈ λ_{∗}(φ) ⊗ λ_{∗}(ψ).

If we use a bit more multilinear algebra, we can come up with the following conceptually more pleasing proof of Theorem 7.

Second proof of Theorem 7. Adopting geometric language, φ corresponds to a bilinear space K ⊗_{𝔬} M and ψ corresponds to a space K ⊗_{𝔬} N with bilinear spaces M and N over 𝔬. Hence φ ⊗ ψ corresponds to the space

(K ⊗_{𝔬} M) ⊗_{K} (K ⊗_{𝔬} N) ≅ K ⊗_{𝔬} (M ⊗_{𝔬} N)

over K. Now the free bilinear module M ⊗_{𝔬} N is again nondegenerate (cf. e.g. [MH, I §5]). Consequently φ ⊗ ψ has good reduction with respect to λ, and λ_{∗}(φ ⊗ ψ) can be represented by the space L ⊗_{λ} (M ⊗_{𝔬} N), obtained from the space M ⊗_{𝔬} N by base extension to L using the homomorphism λ|𝔬 : 𝔬 → L. Now L ⊗_{λ} (M ⊗_{𝔬} N) ≅ (L ⊗_{λ} M) ⊗_{L} (L ⊗_{λ} N). In other words, λ_{∗}(φ ⊗ ψ) ≅ λ_{∗}(φ) ⊗ λ_{∗}(ψ), since λ_{∗}(φ) ˆ= L ⊗_{λ} M, λ_{∗}(ψ) ˆ= L ⊗_{λ} N.

§4 Generic Splitting in Characteristic ≠ 2

In this section we outline an important application area of the specialization theory developed in §3, namely the theory of generic splitting of bilinear forms. Many proofs will only be given in §7, after also having developed a specialization theory of quadratic forms.

Let k be a field and let φ be a form over k, which is just as before understood to be a nondegenerate symmetric bilinear form over k. Our starting point is the following simple

Observation. Let K and L be fields, containing k, and let λ: K → L ∪ ∞ be a place over k, i.e. with λ(c) = c for all c ∈ k.

(a) Then φ⊗K has good reduction with respect to λ and λ_{∗}(φ⊗K) ≈ φ⊗L.^{5} Indeed, if φ ≅ (a_{ij}) with a_{ij} ∈ k, det(a_{ij}) ≠ 0, then also φ ⊗ K ≅ (a_{ij}), and this is a unimodular representation of φ with respect to λ, since k is contained in the valuation ring 𝔬 of λ. So λ_{∗}(φ ⊗ K) ≈ (λ(a_{ij})) = (a_{ij}) and since this symmetric matrix is now considered over L, we conclude that λ_{∗}(φ ⊗ K) ≈ φ ⊗ L.

(b) Suppose that φ ⊗ K has kernel form φ_{1} and Witt index r_{1}, i.e.

φ ⊗ K ≈ φ_{1} ⊥ r_{1} × H,

where H denotes from now on the hyperbolic “plane”^{6} ({0 \atop 1} {1 \atop 0}). According to the corollary of Theorem 4 in §3, the form φ_{1} has good reduction with respect to λ. Therefore, applying λ_{∗} yields

φ ⊗ L ≈ λ_{∗}(φ_{1}) ⊥ r_{1} × H.

Hence we conclude that ind(φ ⊗ L) ≥ ind(φ ⊗ K) and that

λ_{∗}(ker(φ ⊗ K)) ∼ ker(φ ⊗ L).

(Recall the terminology of §2, Definition 1.)

Definition 1. We call two fields K ⊃ k, L ⊃ k over k specialization equivalent, or just equivalent, if there exist places λ: K → L ∪ ∞ and μ: L → K ∪ ∞ over k. We then write K ∼_{k} L.

5 φ ⊗ K is the form φ, considered over K instead of over k. If E is a bilinear space over k corresponding to φ, then K ⊗_{k} E – as described in §3 – is a bilinear space corresponding to φ ⊗ K.

6 We do not make a notational distinction between the forms ( {0 \atop 1} {1 \atop 0} ) over the different fields occurring here.

The following conclusions can be drawn immediately from our observation:

Remark 1. If K ∼_{k} L, then every form φ over k satisfies:

(1) ind(φ ⊗ K) = ind(φ ⊗ L).

(2) ker(φ⊗K) has good reduction with respect to every place λ from K to L and λ_{∗}(ker(φ ⊗ K)) ≅ ker(φ ⊗ L). (Note that “≅” holds, not just “≈”!)

From a technical point of view, it is a good idea to treat the following special case of Definition 1 separately:

Definition 2. We call a field extension K ⊃ k inessential if there exists a place λ: K → k ∪ ∞ over k.

Obviously this just means that K is equivalent to k over k. In this case, Remark 1 becomes:

Remark 2. If K is an inessential extension of k, then every form φ over k satisfies:

(1) ind(φ ⊗ K) = ind(φ),

(2) ker(φ ⊗ K) = ker(φ) ⊗ K.

We will see that the forms φ and φ ⊗ K exhibit the “same” splitting behaviour with respect to an inessential extension K/k in an even broader sense (see Scholium 4 below).

Let us return to the general observation above. It gives rise to the following

Problem. Let φ be a form over k, dim φ = n ≥ 2, and let t be an integer with 1 ≤ t ≤ \frac{n}{2}. Does there exist a field extension K ⊃ k which is “generic with respect to splitting off t hyperbolic planes”?, i.e. with the following properties:

(a) ind(φ ⊗ K) ≥ t.

(b) If L is a field extension of k with ind(φ⊗L) ≥ t, then there exists a place from K to L over k.

We first address this problem for the case t = 1. As before, let φ be a form over a field k.

Definition 3. An extension field K ⊃ k is called a generic zero field of φ if the following conditions hold:

(a) φ ⊗ K is isotropic.

(b) For every field extension k ↪ L with φ ⊗ L isotropic, there exists a place λ: K → L ∪ ∞ over k.

Note that if φ is isotropic, then the field k is of course itself a generic zero field of φ.

In the rest of this section, we assume that char k ≠ 2. Now φ can also be viewed as a quadratic form,^{7} φ(x): = φ(x, x). We define a field extension k(φ) of k, which is a priori suspected to be a generic zero field of φ.

Definition 4. Let dim φ ≥ 3 or let dim φ = 2 and φ ≅ H̸. Let k(φ) denote the function field of the affine quadric φ(X_{1}, …, X_{n}) = 0 (where n = dim φ), i.e. the quotient field of the ring A: = k[X_{1}, …, X_{n}]/(φ(X_{1}, …, X_{n})) with indeterminates X_{1}, …, X_{n}.

Observe that the polynomial φ(X_{1}, …, X_{n}) is irreducible. To prove this we may suppose that φ is diagonalised, φ = ⟨a_{1}, a_{2}, …, a_{n}⟩. The polynomial a_{1}X_{1}^{2} + a_{2}X_{2}^{2} + … + a_{n}X_{n}^{2} is clearly not a product of two linear forms (since char k ≠ 2).

If φ = H, then φ(X_{1}, X_{2}) = X_{1}X_{2}. On formal grounds we then set k(φ) = k(t) for an indeterminate t.

Let x_{1}, …, x_{n} be the images of the indeterminates X_{1}, …, X_{n} in A. Then φ(x_{1}, …, x_{n}) = 0. Hence φ ⊗ k(φ) is isotropic. (This is also true for φ ≅ H.) On top of that we have the following important

Theorem 1. Let dim φ ≥ 2. Then k(φ) is a generic zero field of φ.

Note that the assumption dim φ ≥ 2 is necessary for the existence of a generic zero field, since forms of dimension ≤ 1 are never isotropic.

Theorem 1 can already be found in [K_{5}]. We will wait until §9 to prove it in the framework of a generic splitting theory of quadratic forms.

It is clear from above that every other generic zero field K of φ over k is equivalent to k(φ). It is unknown which bilinear forms possess generic zero fields in characteristic 2.

Given an arbitrary form φ over k, we now construct a tower of fields (K_{r} | 0 ≤ r ≤ h) together with anisotropic forms φ_{r} over K_{r} and numbers i_{r} ∈ ℕ_{0} as follows: choose K_{0} to be any inessential extension of the field k.^{8} Let φ_{0} be the kernel form of φ ⊗ K_{0} and i_{0} the Witt index of φ. Then

φ ⊗ K_{0} ≅ φ_{0} ⊥ i_{0} × H.

If dim φ_{0} ≤ 1, then Stop! Otherwise choose a generic zero field K_{1} ⊃ K_{0} of φ_{0}. Let φ_{1} be the kernel form of φ_{0} ⊗ K_{1} and i_{1} the Witt index of φ_{0} ⊗ K_{1}. Then

φ_{0} ⊗ K_{1} ≅ φ_{1} ⊥ i_{1} × H.

If dim φ_{1} ≤ 1, then Stop! Otherwise choose a generic zero field K_{2} ⊃ K_{1} of φ_{1}. Let φ_{2} be the kernel form of φ_{1} ⊗ K_{2} and i_{2} the Witt index of φ_{1} ⊗ K_{2}. Then

φ_{1} ⊗ K_{2} ≅ φ_{2} ⊥ i_{2} × H,

etc.

7 In keeping with our earlier conventions, it would perhaps be better to write φ(x) =

\frac{1}{2} φ(x, x). However, the factor \frac{1}{2} is not important for now. 8 In earlier works (especially [K_{5}]) K_{0} was always chosen to be k. From a technical

point of view it is favourable to allow K_{0} to be an inessential extension of k, just as in [KR].

The construction halts after h ≤ [\frac{dim φ}{2}] steps with a Witt decomposition

φ_{h−1} ⊗ K_{h} ≅ φ_{h} ⊥ i_{h} × H,

dim φ_{h} ≤ 1.

Definition 5. (K_{r} | 0 ≤ r ≤ h) is called a generic splitting tower of φ. The number h is called the height of φ, denoted h = h(φ). The number i_{r} is called the r-th higher index of φ and the form φ_{r} the r-th higher kernel form of φ.

Remark. Obviously φ_{r} is the kernel form of φ ⊗ K_{r} and ind(φ ⊗ K_{r}) = i_{0} + … + i_{r}. Note that h = 0 iff the form φ “splits”, i.e. iff its kernel form is zero or one-dimensional.

We will see that the number h and the sequence (i_{0}, …, i_{h}) are independent of the choice of generic splitting tower and also that the forms φ_{r} are determined uniquely by φ in a precise way. For this the following theorem is useful.

Theorem 2. Let ψ be a form over a field K. Let λ: K → L ∪ ∞ be a place, such that ψ has good reduction with respect to λ. Suppose that L (and therefore K) has characteristic ≠ 2. Then λ_{∗}(ψ) is isotropic if and only if λ can be extended to a place μ: K(ψ) → L ∪ ∞.

If λ is a trivial place, i.e. a field extension, then this theorem says once more that K(ψ) is a generic zero field of ψ (Theorem 1).

One direction of the assertion is trivial, just as for Theorem 1: if λ can be extended to a place μ: K(ψ) → L ∪ ∞, then ψ ⊗ K(ψ) has good reduction with respect to μ and μ_{∗}(ψ ⊗ K(ψ)) ≅ λ_{∗}(ψ). Since ψ ⊗ K(ψ) is isotropic, λ_{∗}(ψ) is also isotropic.

The other direction will be established in §9. For a shorter and simpler proof, see [K_{5}] and the books [S], [KS].

Theorem 3 (Corollary of Theorem 2). Let φ be a form over a field k. Let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower of φ with associated higher kernel forms φ_{r} and indices i_{r}. Suppose that φ has good reduction with respect to some place γ: k → L ∪ ∞. Suppose that L (and hence k) has characteristic ≠ 2. Finally let λ: K_{m} → L ∪ ∞ be a place for some m, 0 ≤ m ≤ h which extends γ and which \underline{cannot} be extended to K_{m+1} if m < h. Then φ_{m} has good reduction with respect to λ. The form γ_{∗}(φ) has kernel form λ_{∗}(φ_{m}) and Witt index i_{0} + … + i_{m}.

Proof. There is an isometry

(1)  φ ⊗ K_{m} ≅ φ_{m} ⊥ (i_{0} + … + i_{m}) × H.

The form φ ⊗ K_{m} has good reduction with respect to λ and λ_{∗}(φ ⊗ K_{m}) = γ_{∗}(φ). Using Theorem 4 and Theorem 3 from §3, (1) implies that φ_{m} has good reduction with respect to λ and

(2)  γ_{∗}(φ) ≅ λ_{∗}(φ_{m}) ⊥ (i_{0} + … + i_{m}) × H

(cf. the observation at the beginning of this section). If λ_{∗}(φ_{m}) were isotropic, then we would have m < h, since dim φ_{h} ≤ 1. It would then follow from Theorem 2 that λ can be extended to a place from K_{m+1} to L, contradicting our assumption. Therefore λ_{∗}(φ_{m}) is anisotropic and so (2) is the Witt decomposition of γ_{∗}(φ).

Note that this theorem implies in particular, that any attempt to successively extend the given place λ: k → L ∪ ∞ to a “storey” K_{m} of the generic splitting tower, which is as high as possible, always ends at the same number m.

Theorem 3 gives rise to a number of interesting statements about the splitting behaviour of forms and the extensibility of places.

Scholium 1. Let φ be a form over k and (K_{r} | 0 ≤ r ≤ h) a generic splitting tower of φ with kernel forms φ_{r} and indices i_{r}. Furthermore, let k ⊂ L be a field extension. If we apply Theorem 3 to the trivial place γ: k ↪ L, we get:

(1) Let λ: K_{m} → L ∪ ∞ be a place over k (0 ≤ m ≤ h), which cannot be extended to K_{m+1} in case m < h. Then φ_{m} has good reduction with respect to λ and λ_{∗}(φ_{m}) is the kernel form of φ ⊗ L. The index of φ ⊗ L is i_{0} + … + i_{m}.

(2) If λ′: K_{r} → L ∪ ∞ is a place over k, then r ≤ m and λ′ can be extended to a place μ: K_{m} → L ∪ ∞.

(3) Given a number t with 1 ≤ t ≤ [\frac{dim φ}{2}] = i_{0} + … + i_{h}, let m ∈ ℕ_{0} be minimal with t ≤ i_{0} + … + i_{m}. Then K_{m} is a generic field extension of k with respect to splitting off t hyperbolic planes of φ (in the context of our problem above).

Definition 6. Let φ be a form over k. We call the set of indices ind(φ ⊗ L), where L traverses all field extensions of k, the splitting pattern of φ, denoted by SP(φ).

This definition also makes sense in characteristic 2, and it is a priori clear that SP(φ) consists of at most [\frac{dim φ}{2}] + 1 elements. Usually the elements of SP(φ) are listed in ascending order. If char k ≠ 2 and (i_{r} | 0 ≤ r ≤ h) is the sequence of higher indices of a generic splitting tower (K_{r} | 0 ≤ r ≤ h) of φ, then Scholium 1 shows:

SP(φ) = (i_{0}, i_{0} + i_{1}, i_{0} + i_{1} + i_{2}, …, i_{0} + i_{1} + … + i_{h}).

Since the numbers i_{r} with r > 0 are all positive, it is evident that the height h and the higher indices i_{r} (0 ≤ r ≤ h) are independent of the choice of generic splitting tower of φ. Obviously is i_{0} + … + i_{h} = [\frac{dim φ}{2}].

Scholium 2. Let (K_{r} | 0 ≤ r ≤ h) and (K′_{r} | 0 ≤ r ≤ h) be two generic splitting towers of the form φ over k. Applying Scholium 1 to the field extensions k ⊂ K_{s} and k ⊂ K′_{r}, yields: if λ: K_{r} → K′_{s} ∪ ∞ is a place over k, then r ≤ s and λ can be extended to a place μ: K_{s} → K′_{s} ∪ ∞. The fields K_{s} and K′_{s} are equivalent over k. For every place μ: K_{s} → K′_{s} ∪ ∞, the kernel form φ_{s} of φ ⊗ K_{s} has good reduction with respect to μ and μ_{∗}(φ_{s}) is the kernel form φ′_{s} of φ ⊗ K′_{s}.

Scholium 3. Let φ be a form over k and γ: k → L ∪ ∞ a place such that φ has good reduction with respect to γ. Applying Theorem 3 to the places j ∘ γ: k → L ∪ ∞, being the composites of γ and trivial places j: L ↪ L′, yields:

(1) SP(γ_{∗}(φ)) ⊂ SP(φ).

(2) The higher kernel forms of γ_{∗}(φ) arise from certain higher kernel forms of φ by means of specialization. More precisely: if (K_{r} | 0 ≤ r ≤ h) is a generic splitting tower of φ and (K′_{s} | 0 ≤ s ≤ h′) is a generic splitting tower of γ_{∗}(φ), then h′ ≤ h and for every s with 0 ≤ s ≤ h′ we have

ind(γ_{∗}(φ) ⊗ K′_{s}) = i_{0} + … + i_{m}

for some m ∈ {0, …, h}. The number m is the biggest integer such that γ can be extended to λ: K_{m} → K′_{s} ∪ ∞. The kernel form φ_{m} of φ ⊗ K_{m} has good reduction with respect to every extension λ of this kind, and λ_{∗}(φ_{m}) is the kernel form of γ_{∗}(φ) ⊗ K′_{s}.

(3) If ρ: K_{r} → K′_{s} ∪ ∞ is a place, which extends γ: k → L ∪ ∞, then r ≤ m and ρ can be further extended to a place from K_{m} to K′_{s}.

Scholium 4. Let L/k be an inessential field extension (see Definition 2 above) and φ again a form over k. Let (L_{i} | 0 ≤ i ≤ h) be a generic splitting tower of φ ⊗ L. It is then immediately clear from Definition 5 that this is also a generic splitting tower of φ. Hence SP(φ⊗L) = SP(φ), and the higher kernel forms of φ ⊗ L can also be interpreted as higher kernel forms of φ.

Which strictly increasing sequences (0, j_{1}, j_{2}, …, j_{n}) occur as splitting patterns of anisotropic forms? What do anisotropic forms of given height h look like? These questions are difficult and at the moment the object of intense research. The first efforts towards answering them can be found in [K_{4}], [K_{5}], while more recent ones can be found in [H_{2}], [HR_{1}], [HR_{2}], [Ka] amongst others.

A complete answer is only known in the case h = 1. A form ⟨1, a_{1}⟩ ⊗ … ⊗ ⟨1, a_{d}⟩ (d ≥ 1, a_{i} ∈ k) is called a d-fold Pfister form over k.^{9} If τ is a Pfister form of degree d, then the form τ′, satisfying ⟨1⟩ ⊥ τ′ = τ, is called the pure part of τ.

Theorem 4. An anisotropic form φ over k has height 1 if φ ≅ aτ (dim φ even) or φ ≅ aτ′ (dim φ odd), with a ∈ k^{∗} and τ an anisotropic Pfister form of degree d ≥ 1 in the first case and d ≥ 2 in the second case.

9 The form ⟨1⟩ also counts as a Pfister form, more precisely a 0-fold Pfister form.

Note that therefore SP(φ) = (0, 2^{d−1}) when dim φ is even and SP(φ) = (0, 2^{d−1} − 1) when dim φ is odd. Is k = ℝ for example, then all d ≥ 1, resp. d ≥ 2 occur. One can take for instance φ = 2^{d} × ⟨1⟩, resp. φ = (2^{d} − 1) × ⟨1⟩.

A proof of Theorem 4 can be found in [K_{5}] and the books [S], [KS]. In §20 and §23 we will prove two theorems for fields of arbitrary characteristic, from which Theorem 4 can be obtained in characteristic ≠ 2 (§20, Theorem 5, §23, Theorem 1).

Little is still known about forms of height 2, but the known results are interesting and partly deep, see e.g. [K_{6}], [F], [Ka], [H_{1}], [H_{3}].

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§5 An Elementary Treatise on Quadratic Modules

We want to construct a specialization theory for quadratic forms, similar to the theory in §3 for symmetric bilinear forms. In order to do this we need some definitions and theorems about quadratic modules over rings, and in particular over valuation rings.

Let A be any ring (always commutative, with 1). We recall some fundamental definitions and facts about quadratic forms over A, cf. [MH, Appendix 1].

Definition 1. Let M be an A-module. A quadratic form on M is a function q: M → A, satisfying the following conditions:

(1) q(λx) = λ^{2}q(x) for λ ∈ A, x ∈ M.

(2) The function B_{q}: M × M → A, B_{q}(x, y): = q(x + y) − q(x) − q(y) is a bilinear form on M.

The pair (M, q) is then called a quadratic module over A.

Note that the bilinear form B_{q} is symmetric. Furthermore B_{q}(x, x) = 2q(x) for all x ∈ M. If 2 is a unit in A, 2 ∈ A^{∗}, we can retrieve q from B_{q}. Also, every symmetric bilinear form B on M comes from a quadratic form q in this case, namely q(x) = \frac{1}{2}B(x, x). So, if 2 ∈ A^{∗}, bilinear modules (see §3, Def. 2) and quadratic modules over A are really the same objects.

If 2 ∉ A^{∗}, and 2 is not a zero-divisor in A, we can still identify quadratic forms on an A-module M with special symmetric bilinear forms, namely the forms B with B(x, x) ∈ 2A for all x ∈ M (“even” bilinear forms). However, if 2 is a zero-divisor in A, then quadratic and bilinear modules over A are very different objects.

In what follows, primarily free quadratic modules will play a rôle, i.e. quadratic modules (M, q) for which the A-module is free and always of finite rank. If M is a free A-module with basis e_{1}, …, e_{n} and (a_{ij}) a symmetric (n × n)-matrix, then there exists exactly one quadratic form q on M with q(e_{i}) = a_{ii} and B_{q}(e_{i}, e_{j}) = a_{ij} for i ≠ j (1 ≤ i, j ≤ n), namely

q (∑_{i=1}^{n} x_{i}e_{i}) = ∑_{i=1}^{n} a_{ii}x_{i}^{2} + ∑_{1≤i<j≤n} a_{ij}x_{i}x_{j}.

We denote this quadratic module (M, q) by a symmetric matrix in square brackets, (M, q) = [a_{ij}], and call [a_{ij}] the value matrix of q with respect to the basis e_{1}, …, e_{n}. If (a_{ij}) is a diagonal matrix with coefficients a_{1}, …, a_{n}, then we write (M, q) = [a_{1}, …, a_{n}].

Definition 2. Let (M_{1}, q_{1}) and (M_{2}, q_{2}) be quadratic A-modules. The orthogonal sum (M_{1}, q_{1}) ⊥ (M_{2}, q_{2}) is the quadratic A-module

(M_{1} ⊕ M_{2}, q_{1} ⊥ q_{2}),

consisting of the direct sum M_{1} ⊕ M_{2} and the form

(q_{1} ⊥ q_{2})(x_{1} + x_{2}): = q_{1}(x_{1}) + q_{2}(x_{2})

for x_{1} ∈ M_{1}, x_{2} ∈ M_{2}.

If (M_{1}, q_{1}) = [A_{1}] and (M_{2}, q_{2}) = [A_{2}] are free quadratic modules with corresponding symmetric matrices A_{1}, A_{2}, then

(M_{1}, q_{1}) ⊥ (M_{2}, q_{2}) = [{A_{1} \atop 0} {0 \atop A_{2}} ] .

Now it is also clear how to construct a multiple orthogonal sum (M_{1}, q_{1}) ⊥ … ⊥ (M_{r}, q_{r}). In particular we have for elements a_{1}, …, a_{r} ∈ A that

[a_{1}] ⊥ … ⊥ [a_{r}] = [a_{1}, …, a_{r}].

Let (M, q) be a quadratic 𝔬-module and suppose that M_{1} and M_{2} are submodules of the 𝔬-module M, then we write M = M_{1} ⊥ M_{2}, when B_{q}(M_{1}, M_{2}) = 0 (“internal” orthogonal sum, in contrast with the “external” orthogonal sum of Definition 2). Clearly (M, q) ≅ (M_{1}, q|M_{1}) ⊥ (M_{2}, q|M_{2}) in this case.

If β: M × M → A is a – not necessarily symmetric – bilinear form, then q(x): = β(x, x) is a quadratic form on M. If M is free, one can easily verify that every quadratic form on M is of this form. Furthermore, two bilinear forms β, β′ give rise to the same quadratic form q exactly when β − β′ = γ is an alternating bilinear form: γ(x, x) = 0 for all x ∈ M, and hence γ(x, y) = −γ(y, x) for all x, y ∈ M.

Suppose now that (M_{1}, B_{1}) is a free bilinear module (cf. §3) and that (M_{2}, q_{2}) is a free quadratic module. We equip the free module M: = M_{1}⊗_{A}M_{2} with a quadratic form q as follows: first we choose a bilinear form β_{2} on M_{2} with q_{2}(x) = β_{2}(x, x) for all x ∈ M_{2}. Next we form the tensor product β: = B_{1} ⊗ β_{2}: M × M → A of the bilinear forms B_{1}, β_{2}. This bilinear form β is characterized by

β(x_{1} ⊗ x_{2}, y_{1} ⊗ y_{2}) = B_{1}(x_{1}, y_{1})β_{2}(x_{2}, y_{2})

for x_{1}, y_{1} ∈ M_{1} and x_{2}, y_{2} ∈ M_{2} (cf. [Bo_{1}, §1, No.9]). Finally we let q(x): = β(x, x) for x ∈ M_{1} ⊗ M_{2}. This quadratic form q is independent of the choice of the bilinear form β_{2}, since if γ_{2} is an alternating bilinear form on M_{2}, then B_{1} ⊗ γ_{2} is an alternating bilinear form on M.

§5 An Elementary Treatise on Quadratic Modules 31

Definition 3. We call q the tensor product of the symmetric bilinear form B_{1} and the quadratic form q_{2}, denoted q = B_{1} ⊗ q_{2}, and call the quadratic module (M, q) the tensor product of the bilinear module (M_{1}, B_{1}) and the quadratic module (M_{2}, q_{2}), (M, q) = (M_{1}, B_{1}) ⊗ (M_{2}, q_{2}).

The quadratic form q = B_{1} ⊗ q_{2} is characterized by B_{q} = B_{1} ⊗ B_{q_{2}} and

q(x_{1} ⊗ x_{2}) = B_{1}(x_{1}, x_{1}) q_{2}(x_{2})

for x_{1} ∈ M_{1}, x_{2} ∈ M_{2}. For a one-dimensional bilinear module ⟨c⟩ and a quadratic free module (M, q) there is a natural isometry ⟨c⟩ ⊗ (M, q) ≅ (M, cq). In particular is for a symmetric (n × n)-matrix A

⟨c⟩ ⊗ [A] ≅ [cA].

Later on we will often denote a quadratic module (M, q) by the single letter M and an (always symmetric) bilinear module (E, B) by the single letter E. The tensor product E ⊗ M is clearly additive in both arguments,

\begin{aligned}(E_{1} ⊥ E_{2}) ⊗ M & ≅ (E_{1} ⊗ M) ⊥ (E_{2} ⊗ M) \\ E ⊗ (M_{1} ⊥ M_{2}) & ≅ (E ⊗ M_{1}) ⊥ (E ⊗ M_{2}). \end{aligned}

Consequently we have for example (a_{i}, b_{j} ∈ A):

⟨a_{1}, …, a_{r}⟩ ⊗ [b_{1}, …, b_{s}] ≅ [a_{1}b_{1}, a_{1}b_{2}, …, a_{r}b_{s}].

Let M = (M, q) be a free quadratic A-module and α: A → C a ring homomorphism. We associate to M a quadratic C-module M′ = (M′, q′) as follows: The C-module M′ is the tensor product C ⊗_{A} M, formed by means of α. Choose a bilinear module β: M × M → A with q(x) = β(x, x) for all x ∈ M. Let β′: M′ × M′ → C be the bilinear form over C associated to β, in other words

(∗)  β′(c ⊗ x, d ⊗ y) = cd α(β(x, y))

for x, y ∈ M, c, d ∈ C. Let q′(u): = β′(u, u) for u ∈ M′. This quadratic form q′ is independent of the choice of β′, since if γ is an alternating bilinear form on M, then the associated C-bilinear form γ′ on M′ is again alternating. The form q′ can be characterized as follows:

q′(c ⊗ x) = c^{2}α(q(x)), B_{q′}(c ⊗ x, d ⊗ y) = cd α(B_{q}(x, y)),

for x ∈ M, y ∈ M, c ∈ C, d ∈ C.

Definition 4. We say that the quadratic C-module (M′, q′) arises from M = (M, q) by means of a base extension determined by α, and denote M′ by C ⊗_{A} M or by C ⊗_{α} M or, even more precisely, by C ⊗_{A,α} M. We also use the notation q′ = q_{C}.

If M is given by a symmetric matrix, M = [a_{ij}] then C ⊗_{A} M = [α(a_{ij})].

Given a bilinear A-module M = (M, B), we similarly define a bilinear C-module C ⊗_{A} M = (C ⊗_{A} M, B_{C}), where B_{C} is determined in the obvious way by B, cf. (∗) above.

If M = (M, q) is a quadratic module over A and N a submodule of M, we also interpret N as a quadratic module, N = (N, q|N) (quadratic submodule). Furthermore we denote by N^{⊥} the submodule of M consisting of all elements x ∈ M with B_{q}(x, y) = 0 for all y ∈ N. Of course, we also interpret N^{⊥} as a quadratic module. In particular we can look at the quadratic submodule M^{⊥} of M. If M^{⊥} is free of finite rank r, then M^{⊥} has the form [a_{1}, …, a_{r}] with elements a_{i} ∈ A.

Later on we will frequently use the following elementary lemma.

Lemma 1 [MH, p.5]. Let M = (M, B) be a bilinear A-module and let N be a submodule of M. Suppose that the bilinear form B is nondegenerate on N, i.e. the homomorphism x ↦ B(x, −)|N from N to the dual module

N^{∨} = Hom_{A}(N, A) is an isomorphism. Then M = N ⊥ N^{⊥}.

Let M = (M, q) be a quadratic module and let N be a submodule of M such that the bilinear form B_{q} is nondegenerate on N, then according to the lemma we also have that M = N ⊥ N^{⊥}.

To finish this section, we briefly examine free hyperbolic modules, being quadratic modules of the form r × [{0 \atop 1} {1 \atop 0}] with r ∈ ℕ_{0}, in other words direct sums of r copies of the quadratic module [{0 \atop 1} {1 \atop 0}]. \{When r = 0, the zero module is meant.}

Lemma 2. Let (M, q) be a quadratic A-module whose associated bilinear form B_{q} is nondegenerate. Let U be a submodule of M with q|U = 0. Suppose that U is free of rank r and that it is a direct summand of the A-module M. Then there exists a submodule N ⊃ U of M with N ≅ r × [{0 \atop 1} {1 \atop 0}] and M = N ⊥ N^{⊥}.

Proof. (cf. [Ba, p.13 f]). We saw already in §3 (just after Definition 5) that there exists an exact sequence of A-modules

0 ⟶ U^{⊥} ⟶ M ⟶^{α} U^{∨} ⟶ 0,

with U^{∨}: = Hom_{A}(U, A), where α maps an element z ∈ M to the linear form

B(z, −)|U on U. Since U^{∨} is free, this sequence splits. Choose a submodule

V of M with M = U^{⊥} ⊕ V. Then α|V: V → U^{∨} is an isomorphism. The modules U and V are therefore in perfect duality with respect to the pairing U × V → A, (x, y) ↦ B_{q}(x, y).

Now choose a – not necessarily symmetric – bilinear form β: V × V → A with β(x, x) = q(x) for x ∈ V. We then have an A-linear map φ: V → U such that β(v, x) = B_{q}(v, φ(x)) for v ∈ V, x ∈ V. Therefore

q(x − φ(x)) = q(x) − B_{q}(x, φ(x)) = q(x) − β(x, x) = 0

for every x ∈ V. Let W: = {x − φ(x) | x ∈ V }. Since

B_{q}(u, x − φ(x)) = B_{q}(u, x)

for all u ∈ U and x ∈ V, the modules W and U are also in perfect duality with respect to B_{q}. Furthermore is q|W = 0. Choose a basis e_{1}, …, e_{r} for U and let f_{1}, …, f_{r} be the dual basis of W with respect to B_{q}, i.e. B_{q}(e_{i}, f_{j}) = δ_{ij}. Then N: = U ⊕ W can be written as

N = ⊥_{i=1}^{r}(Ae_{i} + Af_{i}) ≅ r × [{0 \atop 1} {1 \atop 0}] .

An appeal to Lemma 1 yields M = N ⊥ N^{⊥}.

Given a quadratic A-module M = (M, q), we denote (M, −q) by −M, as is already our practice for bilinear spaces.

Theorem. Let M = (M, q) be a quadratic A-module. Suppose that the bilinear form B_{q} is nondegenerate and that M is free of rank r. Then

M ⊥ (−M) ≅ r × [{0 \atop 1} {1 \atop 0}] .

Proof. Let (E, q̃): = (M, q) ⊥ (M, −q). We interpret the module E = M ⊕M as the cartesian product M × M. The diagonal D: = {(x, x) | x ∈ M} is a free E-submodule of rank r and D is a direct summand of E, for E = D ⊕ (M × {0}). Furthermore q|D = 0. The statement now follows from Lemma 2.

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§6 Quadratic Modules over Valuation Rings

In this section we let 𝔬 be a valuation ring with maximal ideal 𝔪, quotient field K and residue class field k = 𝔬/𝔪. The case 𝔪 = 0, i.e. K = k = 𝔬 is explicitly allowed. We shall present the theorems about quadratic modules over 𝔬 necessary for our specialization theory, and prove most of them.

Lemma 1. Let (M, B) be a free bilinear module over 𝔬, and let N be a submodule of M. The submodule N^{⊥} of M is free with finite basis and is a direct summand of M. Every submodule L of M with M = N^{⊥} ⊕ L is also free with finite basis.

Proof. M/N^{⊥} is torsion free and finitely generated, hence free with finite basis. This follows from the fact that every finitely generated ideal of 𝔬 is principal, cf. [CE, VII, §4]. (We used this already in the proof of Lemma 2, §3.) If \overline{x}_{1}, …, \overline{x}_{r} is a basis of M/N^{⊥} and x_{1}, …, x_{r} are the pre-images of the \overline{x}_{i} in M, then L_{0}: = 𝔬x_{1} + … + 𝔬x_{r} is free with basis x_{1}, …, x_{r} and M = N^{⊥} ⊕ L_{0}. If M = N^{⊥} ⊕ L as well, then L ≅ M/N^{⊥} ≅ L_{0}, so that L is also free with finite basis. Finally, N^{⊥} is torsion free and finitely generated, hence free with finite basis.

In particular, if (M, B) is a free bilinear module over 𝔬, there is an orthogonal decomposition M = M_{0} ⊥ M^{⊥} with B|M^{⊥} × M^{⊥} = 0.

Definition 1. M^{⊥} is called the radical of the bilinear module (M, B). If (M, q) is a free quadratic 𝔬-module, then the radical M^{⊥} of (M, B_{q}), equipped with the quadratic form q|M^{⊥}, is called the quasilinear part of M, and is denoted by QL(M). If M = M^{⊥}, the quadratic module (M, q) is called quasilinear.

Definition 2. Let M be a free 𝔬-module with basis e_{1}, …, e_{n}. We call a vector x ∈ M primitive in M, if for the decomposition x = λ_{1}e_{1} + … + λ_{n}e_{n}

(λ_{i} ∈ 𝔬) the ideal ∑_{1}^{m} λ_{i}𝔬 is equal to 𝔬, i.e. at least one λ_{i} is a unit.

This property is independent of the choice of basis: x is primitive when M/𝔬x is torsion free, hence when 𝔬x is a direct summand of the module M.

Definition 3. We call a quadratic module (M, q) over 𝔬 nondegenerate if it satisfies the following conditions:

(Q0) M is free of finite rank. (Q1) The bilinear form \overline{B}_{q}, induced by B_{q} on M/M^{⊥} in the obvious way,

is nondegenerate. (Q2) q(x) ∈ 𝔬^{∗} for every vector x in M^{⊥}, which is primitive in M^{⊥} (and

hence in M).

If instead of (Q2), the following condition is satisfied (Q2′) QL(M) = 0 or QL(M) ≅ [ε] with ε ∈ 𝔬^{∗}, then we call (M, q) regular. In the special case M^{⊥} = 0, we call (M, q) strictly regular.

Remark. The strictly regular quadratic modules are identical with the “quadratic spaces” over 𝔬 in [K_{4}]. In case 𝔬 = K, the term “nondegenerate” has the same meaning as in [K_{4}]. “Strictly regular” is a neologism, constructed with the purpose of avoiding collision with the confusingly many overlapping terms in the literature.

From a technical point of view, Definition 3 is the core definition of this book. The idea behind it is, that the term “nondegenerate” captures a possibly large class of quadratic 𝔬-modules, which work well in the specialization theory (see §7 and following). {We will settle upon the agreement that a quadratic K-module E has “good reduction with respect to 𝔬” if E ≅ K ⊗_{𝔬} M, where M is a nondegenerate quadratic 𝔬-module, see §7, Def.1.} The requirements (Q0) and (Q1) are obvious, but (Q2) and (Q2′) deserve some explanation.

If char K ≠ 2, i.e. 2 ≠ 0 in 𝔬, then q|M^{⊥} = 0 and the requirement (Q2) implies that M^{⊥} = 0, hence implies – in conjunction with (Q0) and (Q1) – strict regularity. If 2 ∈ 𝔬^{∗}, then the nondegenerate quadratic 𝔬-modules are the same objects as the nondegenerate bilinear 𝔬-modules in the sense of our earlier definition.

Suppose now that char K = 2. The condition M^{⊥} = 0, in other words strict regularity, is very natural, but too limited for applications. Indeed, if M^{⊥} = 0, then the bilinear module (M, B_{q}) is nondegenerate and we have B_{q}(x, x) = 2q(x) = 0 for every x ∈ M. This implies that M has even dimension, as is well-known. (To prove this, consider the vector space K ⊗_{𝔬} M.) So if we insist on using strict regularity, we can only deal with quadratic forms of even dimension.

On the other hand, property (Q2) has an annoying defect: it is not always preserved under a basis extension. If 𝔬′ ⊃ 𝔬 is another valuation ring, whose maximal ideal 𝔪′ lies over 𝔪, i.e. 𝔪′ ∩𝔬 = 𝔪, and if M is nondegenerate, then 𝔬′ ⊗_{𝔬} M can be degenerate.

However, if M satisfies (Q2′), this clearly cannot happen. Therefore we will limit ourselves later (from §9 onwards) in some important cases to regular modules.

The arguments will be clearer however, when we allow arbitrary nondegenerate quadratic modules as long as possible, and a lot of results in this generality are definitely important.

Our use of the terms “regular” and “nondegenerate” finds its justification in the requirements of the specialization theory presented in this book.

Regular quadratic modules over valuation rings and fields (in the sense of Definition 3) are just those quadratic modules, for which the generic splitting theory in particular functions in a “regular way”, as we are used to in the absence of characteristic 2 (cf. §4), see §9 below. The nondegenerate quadratic modules are those ones, for which the generic splitting theory still can get somewhere in a sensible way, see §10 and §12 below.

The author is aware of the fact that in other areas (number theory in particular) a different terminology is used. For example, Martin Kneser calls our regular modules “half regular” and our strictly regular modules “regular” in his lecture notes [Kn_{3}], and has a good reason to do so. In the literature, the word “nondegenerate” is almost exclusively used for the more restrictive class of quadratic modules, which we call “strictly regular”, see Bourbaki [Bo_{1}, §3, No. 4] in particular. In our context, however, it would be a little bit silly to give the label “degenerate” to every quadratic module, which is not strictly regular.

We need some formulas, related to quadratic modules of rank 2.

Lemma 2. Let α ∈ 𝔬, β ∈ 𝔬, λ ∈ 𝔬^{∗}. Then

(1)  [{α \atop λ} {λ \atop β} ] ≅ [{α \atop 1} {1 \atop λ^{−2}β} ] ,

and furthermore

(2)  ⟨λ⟩ ⊗ [{α \atop 1} {1 \atop β} ] ≅ [{λα \atop 1} {1 \atop λ^{−1}β} ] .

Finally, if α ∈ 𝔬^{∗}, β ∈ 𝔬 then

(3)  [{α \atop 1} {1 \atop β} ] ≅ ⟨α⟩ ⊗ [{1 \atop 1} {1 \atop αβ} ] .

Proof. Let (M, q) be a free quadratic module with basis e, f and associated value matrix [{α \atop λ} {λ \atop β}]. Then e, λ^{−1}f is also a basis of M, whose associated

value matrix equals [{α \atop 1} {1 \atop λ^{−2}β} ]. This settles (1). Furthermore ⟨λ⟩ ⊗ [{α \atop 1} {1 \atop β} ] is

determined by the quadratic module (M, λq), and so ⟨λ⟩⊗[{α \atop 1} {1 \atop β} ] ≅ [{λα \atop λ} {λ \atop λβ} ]. Applying (1) to this, results in (2). Clearly (3) is a special case of (2).

Theorem 1. Let M = (M, q) be a nondegenerate quadratic module over 𝔬.

(a) Then M is an orthogonal sum of modules [{α \atop 1} {1 \atop β} ] with 1 − 4αβ ∈ 𝔬^{∗} and modules [ε] with ε ∈ 𝔬^{∗}.

(b) If M is regular and dim M even (recall that dim M denotes the rank of the free module M), then M is strictly regular and equal to an orthogonal sum of modules [{α \atop 1} {1 \atop β} ] with 1 − 4αβ ∈ 𝔬^{∗}.

Proof. First suppose that 2 ∈ 𝔬^{∗}. Then (M, B_{q}) is a bilinear space. Hence (M, B_{q}) has an orthogonal basis, as is well-known (cf. [MH, I, Cor.3.4]), and so

(M, B_{q}) ≅ ⟨ε_{1}, …, ε_{n}⟩.

Therefore (M, q) ≅ [\frac{ε_{1}}{2} , …, \frac{ε_{n}}{2} ]. For a binary quadratic module N = [a, b] with units a, b ∈ 𝔬^{∗}, we have

[a, b] ≅ [{a \atop 2a} {2a \atop a + b}] ,

since, if e, f is a basis of the module N with value matrix [{a \atop 0} {0 \atop b}], then e, e + f

is a basis with value matrix [{a \atop 2a} {2a \atop a+b}]. Lemma (2), formula (1) then yields

[a, b] ≅ [{a \atop 1} {1 \atop (2a)^{−1}(a + b)}] .

Hence all the assertions of the theorem are clear when 2 ∈ 𝔬^{∗}.

Next suppose that 2 ∈ 𝔪. The quasilinear part M^{⊥} is of the form [ε_{1}, …, ε_{r}] with ε_{i} ∈ 𝔬^{∗}. Let N be a module complement of M^{⊥} in M. Then (N, q|N) is strictly regular. We will show, by induction on dim N, that N is the orthogonal sum of binary modules [{α \atop 1} {1 \atop β} ].

If N = 0, nothing has to be done. So suppose that N ≠ 0. We choose a primitive vector e in N. Since the bilinear form B_{q}|N × N is nondegenerate, there exists a vector f ∈ N with B_{q}(e, f) = 1. Let α = q(e), β = q(f). The determinant of the matrix ({2α \atop 1} {1 \atop 2β} ) is the unit 1 − 4αβ. Hence the module 𝔬e + 𝔬f is free with basis e, f. Using Lemma 1 from §5, we get an orthogonal decomposition N = (𝔬e + 𝔬f) ⊥ N_{1}. Now 𝔬e + 𝔬f ≅ [{α \atop 1} {1 \atop β} ], and by our induction hypothesis, N_{1} is also an orthogonal sum of binary quadratic modules of this form.

This establishes part (a) of the proof. In particular, N has even dimension. If M is regular, then r ≤ 1. Is in addition dim M even, then r = 0, in other words, M has to be strictly regular.

If M is a free 𝔬-module, we interpret M as a subset of the K-vecor space K ⊗_{𝔬} M, by identifying x ∈ M with 1 ⊗ x. Then K ⊗_{𝔬} M = KM.

Definition 4. A quadratic 𝔬-module (M, q) is called isotropic if there exists a vector x ≠ 0 in M with q(x) = 0. Otherwise (M, q) is called anisotropic.

Remark. In this definition of “isotropic”, we proceed in a different way, compared to §3, Definition 6, since the bilinear form B_{q} can be degenerate. This would cause trouble over a local ring instead of over 𝔬.

Lemma 3. Suppose that (M, q) is a quadratic 𝔬-module, and that M is free of finite rank over 𝔬. The following assertions are equivalent:

(a) (M, q) is isotropic.

(b) (K ⊗_{𝔬} M, q_{K}) is isotropic.

(c) The module M has a direct summand V ≠ 0 with q|V = 0.

Proof. The implications (a) ⇒ (b) and (c) ⇒ (a) are trivial. (b) ⇒ (c): Let E = K ⊗_{𝔬} M = KM. Now q = q_{K}|M. The K-vector space E contains a subspace W ≠ 0 with q_{K}|W = 0. Let V: = W ∩ M. The 𝔬-module V is a direct summand of M, since M/V is torsion free and finitely generated, hence free. We have KV = W. Therefore certainly V ≠ 0 and q|V = 0.

Definition 5. Let (M, q) be a quadratic 𝔬-module. A pair of vectors e, f in M is called hyperbolic if q(e) = q(f) = 0 and B_{q}(e, f) = 1.

Note that according to §5, Lemma 1, we then have that

M = (𝔬e + 𝔬f) ⊥ N ≅ [{0 \atop 1} {1 \atop 0}] ⊥ N

where N is a quadratic submodule of M, namely N = (𝔬e + 𝔬f)^{⊥}.

Lemma 4. Let (M, q) be a nondegenerate quadratic 𝔬-module, and let e be a primitive vector in M with q(e) = 0. Then e can be completed to a hyperbolic vector pair e, f.

Proof. We choose a decomposition M = N ⊥ M^{⊥} and write e = x + y with x ∈ N, y ∈ M^{⊥}. Suppose for the sake of contradiction that the vector x is not primitive in N, hence not primitive in M. Then y is primitive in M. According to condition (Q2), we then have q(y) ∈ 𝔬^{∗}. Hence also q(x) = −q(y) ∈ 𝔬^{∗}. Therefore x has to be primitive, contradiction.

Hence x is primitive in N. Since B_{q} is nondegenerate on N, there exists an element z ∈ N with B_{q}(x, z) = 1. We also have B_{q}(e, z) = 1. Clearly f: = z − q(z)e completes the vector e to a hyperbolic pair.

Theorem 2 (Cancellation Theorem). Let M and N be free quadratic 𝔬modules. Let G be a strictly regular quadratic 𝔬-module, and suppose that G ⊥ M ≅ G ⊥ N. Then M ≅ N.

For the proof we refer to [K_{3}], in which such a cancellation theorem is proved over local rings. An even more general theorem can be found in Kneser’s works [Kn_{2}], [Kn_{3}, Ergänzung zu Kap. I]. A very accessible source for many aspects of the theory of quadratic forms over local rings is the book [Ba] of R. Baeza. Admittedly, Baeza only treats strictly regular quadratic modules, which he calls “quadratic spaces”.

In the field case 𝔬 = K, Theorem 2 was already demonstrated for 2 = 0 by Arf [A] and for 2 ≠ 0 – as is very well-known – by Witt [W]. A nice discussion of the cancellation problem over fields, with a view towards the theory over local rings, can again be found in Kneser’s work [Kn_{1}].

Theorem 3. Let M be a nondegenerate quadratic module over 𝔬. There exists a decomposition

(∗)  M ≅ M_{0} ⊥ r × [{0 \atop 1} {1 \atop 0}]

with M_{0} nondegenerate and anisotropic, r ≥ 0. The number r ∈ ℕ_{0} and the isometry class of M_{0} are uniquely determined by M.

Proof. Lemma 3 and Lemma 4 immediately imply the existence of a decomposition (∗). Suppose now that

M ≅ M′_{0} ⊥ r′ × [{0 \atop 1} {1 \atop 0}]

is another decomposition. Suppose without loss of generality that r ≤ r′. Then Theorem 2 implies that

M_{0} ≅ M′_{0} ⊥ (r′ − r) × [{0 \atop 1} {1 \atop 0}] .

Since M_{0} is anisotropic, we must have r = r′.

In the field case 𝔬 = K, this theorem can again be found back in Arf [A] for 2 = 0 and in Witt [W] for 2 ≠ 0.

Definition 6. We call a decomposition (∗), as in Theorem 3, a Witt decomposition of M. We call M_{0} the kernel module of the nondegenerate quadratic module M and r the Witt index of M and write M_{0} = ker(M), r = ind(M).

Furthermore, we call two nondegenerate quadratic modules M and N over 𝔬 Witt equivalent, denoted by M ∼ N, when ker(M) ≅ ker(N). We denote the Witt class of M, i.e. the equivalence class of M with respect to ∼, by {M}.

It is now tempting to define an addition of Witt classes {M}, {N} of two nondegenerate quadratic modules M, N over 𝔬 in the same way as we did this for nondegenerate bilinear modules over fields in §2, namely {M} + {N}: = {M ⊥ N}.

Although the orthogonal sum of two nondegenerate quadratic modules could be degenerate, the addition makes sense if one of the modules M, N is strictly regular, as we will show now.

Proposition. Let M, M′ be strictly regular quadratic 𝔬-modules with M ∼ M′, and let N, N′ be nondegenerate quadratic 𝔬-modules with N ∼ N′. Then the modules M ⊥ N and M′ ⊥ N′ are nondegenerate and M ⊥ N ∼ M′ ⊥ N′.

Proof. It follows immediately from Definition 3 that M ⊥ N and M′ ⊥ N′ are nondegenerate. Suppose without loss of generality that dim M′ ≥ dim M. Then M′ ≅ M ⊥ s × [{0 \atop 1} {1 \atop 0}] for an s ≥ 0. Therefore M′ ⊥ N ≅ (M ⊥ N) ⊥ s × [{0 \atop 1} {1 \atop 0}], and so M′ ⊥ N ∼ M ⊥ N. A similar argument shows that M′ ⊥ N ∼ M′ ⊥ N′.

Definition 7a. We denote the set of Witt classes of nondegenerate quadratic 𝔬-modules by W̃q(𝔬). We denote the subset of Witt classes of regular, resp. strictly regular quadratic 𝔬-modules by Wqr(𝔬), resp. Wq(𝔬).

Because of the proposition above, we now have a well-defined “addition” of classes {M} ∈ Wq(𝔬) with classes {N} ∈ W̃q(𝔬),

{M} + {N}: = {M ⊥ N}.

The zero module M = 0 gives rise to a neutral element for this addition, {0} + {N} = {N}, {M} + {0} = {M}. If M and N are strictly regular, then {M} + {N} = {N} + {M} = {M ⊥ N} ∈ Wq(𝔬). Therefore Wq(𝔬) is an abelian semigroup with respect to this sum. The theorem at the end of §5 shows that Wq(𝔬) is in fact an abelian group, since if (M, q) is a strictly regular quadratic 𝔬-module, then {(M, q)} + {(M, −q)} = 0.

Definition 7b. We call the abelian group Wq(𝔬) the quadratic Witt group of 𝔬, the set W̃q(𝔬) the quadratic Witt set of 𝔬 and Wqr(𝔬) the regular quadratic Witt set of 𝔬.

The group Wq(𝔬) acts on the set W̃q(𝔬) by means of the addition, introduced above. One can show easily that the isomorphism classes of nondegenerate quasilinear quadratic 𝔬-modules (see Definition 1) form a system of representatives of the orbits of this action. Wqr(𝔬) is a union of orbits.

If 𝔪 = 0, hence 𝔬 = K, then Witt classes of degenerate quadratic modules can be defined without difficulties. We will explain this next.

Definition 8. Let M = (M, q) be a finite dimensional quadratic module over K. The defect δ(M) of M is the set of all x ∈ M^{⊥} with q(x) = 0.

The defect δ(M) is clearly a subspace of M, and M is nondegenerate if and only if δ(M) = 0. The form q induces a quadratic form \overline{q}: M/δ(M) → K on the vector space M/δ(M) in the obvious way: \overline{q}(\overline{x}): = q(x) for x ∈ M, where \overline{x} is the image of x in M/δ(M). The quadratic K-module (M/δ(M), \overline{q}) is clearly nondegenerate.

Definition 9. If δ(M) = 0, i.e. if M is nondegenerate, then we call M = (M, q) a quadratic space over K. In general we denote the quadratic space

(M/δ(M), \overline{q}) over K by M^{∧}, and call M^{∧} the quadratic space associated to M.

We have

M ≅ M^{∧} ⊥ δ(M) ≅ M^{∧} ⊥ s × [0]

with s: = dim δ(M). Furthermore, given a quadratic space E over K, we obviously have

δ(E ⊥ M) = δ(M) and (E ⊥ M)^{∧} ≅ E ⊥ M^{∧} .

Now it is clear, that in the field case Theorem 3 can be expanded as follows:

Theorem 3′. Let M be a finite dimensional quadratic K-module. There exists a decomposition

M ≅ M_{0} ⊥ r × [{0 \atop 1} {1 \atop 0}]

with M^{∧} _{0} anisotropic and r ≥ 0. The number r ∈ ℕ_{0} and the isometry class of M_{0} are uniquely determined by M.

Definition 10. We call M_{0} the kernel module of M and r the Witt index of M, and write M_{0} = ker M, r = ind(M). Just as before, we call two finite dimensional quadratic K-modules M and N Witt equivalent, and write M ∼ N, when ker(M) ≅ ker(N). We denote the Witt equivalence class of M by {M}.

Remark. It may seem more natural to call M and N equivalent when their “kernel spaces” (ker M)^{∧}, (ker N)^{∧} are isometric. Our results in §7 and §11 are a bit stronger if we use the finer equivalence defined above.

We denote the set of Witt classes {M} of finite dimensional quadratic K-modules M by Ŵq(K) and call this the defective quadratic Witt set of K. The abelian group Wq(K) acts on Ŵq(K) in the usual way: {E} + {M}: = {E ⊥ M} for {E} ∈ Wq(K), {M} ∈ Ŵq(K). It even makes sense to add two arbitrary Witt classes together:

{M} + {N}: = {M ⊥ N}.

The result is independent of the choice of representatives M, N, as can be shown by an argument, analogous to the proof of the proposition, preceding Definition 7a. Therefore Ŵq(K) is an abelian semigroup, having Wq(K) as a subgroup.

Caution! If char K = 2, the semigroup Ŵq(K) does not satisfy a cancellation rule. Is for example a ∈ K^{∗}, then [a] ⊥ [a] ≅ [0] ⊥ [a], but it is not so that [a] ∼ [0].

To a field homomorphism α: K → K′ (i.e. a field extension), we can associate a well-defined map α_{∗}: Ŵq(K) → Ŵq(K′), which sends the Witt class {M} to the class {K′ ⊗_{α} M}. This map is a semigroup homomorphism. By restricting α_{∗}, we can find maps from Wq(K) to Wq(K′) and from Wqr(K) to Wqr(K′), but (if char K = 2) in general no map from W̃q(K) to W̃q(K′). This is the reason why we sometimes will extend the Witt set W̃q(K) to the semigroup Ŵq(K).

Let us now return to quadratic modules over arbitrary valuation rings. At the end of §5 we introduced free hyperbolic modules. From now on, we

call those modules simply “hyperbolic”. { In this way, we stay in harmony with the terminology, used in the literature, for valuation rings (or local rings in general), cf. [B_{1}, V §2], [B_{2}].} Hence

Definition 11. A quadratic 𝔬-module M is called hyperbolic if

M ≅ r × [{0 \atop 1} {1 \atop 0}]

for some r ∈ ℕ_{0}.

So, if M is not the zero module, this means that M contains a basis e_{1}, f_{1}, e_{2}, f_{2}, …, e_{r}, f_{r}, where e_{i}, f_{i} are hyperbolic vector pairs, such that 𝔬e_{i}+ 𝔬f_{i} is orthogonal with 𝔬e_{j} + 𝔬f_{j} for i ≠ j. Clearly a hyperbolic module is strictly regular.

Lemma 5. Let M = (M, q) be a strictly regular quadratic 𝔬-module. M is hyperbolic if and only if M contains a submodule V with q|V = 0 and V = V^{⊥}.

Proof. If M is hyperbolic and e_{1}, f_{1}, …, e_{r}, f_{r} is a basis, consisting of pairwise orthogonal hyperbolic vector pairs, then the module V: = 𝔬e_{1} +𝔬e_{2} +…+𝔬e_{r} clearly has the properties q|V = 0 and V = V^{⊥}. Suppose now that M is strictly regular and that V is a submodule satisfying these properties. Since M/V^{⊥} = M/V is torsion free and finitely generated, it is free. Therefore V is a direct summand of the 𝔬-module M. V is torsion free and finitely generated as well, and so also free. The form B_{q} gives rise to an exact sequence

0 ⟶ V^{⊥} ⟶ M ⟶ V^{∨} ⟶ 0

(cf. the proof of §5, Lemma 2), from which we deduce that dim M = dim V + dim V^{⊥} = 2 dim V.

According to §5, Lemma 2 there is an orthogonal decomposition M = N ⊥ N^{⊥} with V ⊂ N and N hyperbolic. The module V^{⊥} built in N instead of M also coincides with V. We conclude that dim N = 2 dim V as well, so that N = M.

Theorem 4. Let M be a strictly regular quadratic 𝔬-module with K ⊗_{𝔬} M hyperbolic. Then M itself is hyperbolic.

Proof. We apply the criterion, given by Lemma 5, two times. As before we interpret M as an 𝔬-submodule of E: = K ⊗_{𝔬} M, i.e. M ⊂ E, E = KM. Now E contains a subspace W with q|W = 0, W^{⊥} = W. The intersection V: = W ∩ M is an 𝔬-submodule of M with q|V = 0 and V^{⊥} = V. (Here V^{⊥} is considered in M.)

Definition 12. The valuation ring 𝔬 is called quadratically henselian if for every γ ∈ 𝔪, there exists a λ ∈ 𝔬 such that λ^{2} − λ = γ.

Remark. One can convince oneself that this means that “Hensel’s Lemma” is satisfied by monic polynomials of degree 2. Every henselian valuation ring is of course quadratically henselian (cf. [Ri, Chap. F], [E, §16]). In Definition 12 we isolated that part of the property “henselian”, which is important for the theory of quadratic forms. Note that, if 𝔬 is quadratically henselian, then every ε ∈ 1 + 4𝔪 is a square in 𝔬, since ε = 1 + 4γ and γ = λ^{2} − λ imply that ε = (1 − 2γ)^{2}.

Lemma 6. Let 𝔬 be quadratically henselian. Let α, β ∈ K and αβ ∈ 𝔪. Then we have that over K:

[{α \atop 1} {1 \atop β} ] ≅ [{0 \atop 1} {1 \atop 0}] .

Proof. According to Lemma 2, formula (3), we have

[{α \atop 1} {1 \atop β} ] ≅ ⟨α⟩ ⊗ [{1 \atop 1} {1 \atop αβ} ] .

There exists a λ ∈ 𝔬 with λ^{2} + λ + αβ = 0. If e, f is a basis of the module [{1 \atop 1} {1 \atop αβ} ], having the indicated value matrix, then q(λe+f) = λ^{2} +λ+αβ = 0.

Therefore [{α \atop 1} {1 \atop β} ] is isotropic. Applying Lemma 4, with 𝔬 = K, gives [{α \atop 1} {1 \atop β} ] ≅

[{0 \atop 1} {1 \atop 0}].

Theorem 5. Let 𝔬 be quadratically henselian. Let (M, q) be a nondegenerate anisotropic quadratic module over 𝔬. Then q(e) is a unit for every vector e ∈ M which is primitive in M.

Proof. We choose a decomposition M = N ⊥ M^{⊥}. Let e ∈ M be primitive. Suppose for the sake of contradiction that q(e) ∈ 𝔪. We write e = x + y with x ∈ N, y ∈ M^{⊥}, and have q(e) = q(x) + q(y).

Case 1: x is primitive in M, hence also in N. Then there exists an f ∈ N with B_{q}(x, f) = 1, since B_{q} is nondegenerate on N. We have B_{q}(e, f) = 1. Therefore

𝔬e + 𝔬f ≅ [{α \atop 1} {1 \atop β} ]

with α = q(e) ∈ 𝔪, β = q(f) ∈ 𝔬. According to Lemma 6, 𝔬e+𝔬f is isotropic. This is a contradiction, since M is anisotropic.

Case 2: x is not primitive in M. Now x = λx_{0} with x_{0} primitive in M, λ ∈ 𝔪. The vector y has to be primitive in M. Since q is nondegenerate, it follows that q(y) ∈ 𝔬^{∗}. We have

q(e) = λ^{2}q(x_{0}) + q(y) ∈ 𝔬^{∗},

contradicting the assumption that q(e) ∈ 𝔪.

We conclude that q(e) ∈ 𝔬^{∗}.

Later on, we will use Lemma 6 and Theorem 5 for a nondegenerate quadratic module M over an arbitrary valuation ring 𝔬, by going from M to the quadratic module M^{h}: = 𝔬^{h} ⊗_{𝔬} M over the henselisation 𝔬^{h} of 𝔬 (see [Ri, Chap.F], [E, §17]). {Remark: One can also form a “quadratic henselisation” 𝔬^{qh} in the obvious way. It would suffice to take 𝔬^{qh} instead of 𝔬^{h}.} In order to do so, the following lemma is important.

Lemma 7. Let M be a nondegenerate quadratic module over 𝔬. Then the quadratic module M^{h} over 𝔬^{h} is also nondegenerate.

Proof. The properties (Q0) and (Q1) from Definition 3 stay valid for every basis extension. Hence we suppose that M satisfies (Q0) and (Q1), so that we only have to consider (Q2). We have (M^{h})^{⊥} = (𝔬^{h} ⊗_{𝔬} M)^{⊥} = 𝔬^{h} ⊗_{𝔬} M^{⊥} = (M^{⊥})^{h}.

The property (Q2) for M says that the quasilinear space k ⊗_{𝔬} M^{⊥} = k ⊗_{𝔬} QL(M) is anisotropic over k. To start with, we can identify k ⊗_{𝔬} M with the quadratic module M/𝔪M over k = 𝔬/𝔪, whose quadratic form \overline{q}: M/𝔪M → k is induced by the quadratic form q: M → 𝔬 in the obvious way. In the same way we then have k ⊗_{𝔬} M^{⊥} = M^{⊥}/𝔪M^{⊥}. A vector x ∈ M^{⊥} is primitive if and only if its image \overline{x} ∈ M^{⊥}/𝔪M^{⊥} is nonzero. (Q2) implies then that \overline{q}(\overline{x}) ≠ 0.

Let 𝔪^{h} denote the maximal ideal of 𝔬^{h}. It is well-known that 𝔪^{h} ∩ 𝔬 = 𝔪 and that the residue class field 𝔬^{h}/𝔪^{h} is canonically isomorphic to 𝔬/𝔪 = k. We identify 𝔬^{h}/𝔪^{h} with k. Then k⊗_{𝔬^{h}}M^{h} = k⊗_{𝔬}M and also k⊗_{𝔬^{h}}QL(M^{h}) = k ⊗_{𝔬} QL(M). Therefore (Q2) is satisfied for M if and only if (Q2) is satisfied for M^{h}.

46 Chapter I. Fundamentals of Specialization Theory

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§7 Weak Specialization

In this section λ: K → L∪∞ is a place, 𝔬 = 𝔬_{λ} is the valuation ring associated to λ, with maximal ideal 𝔪 and residue class field k = 𝔬/𝔪. Given a quadratic module E = (E, q) over K, we want to associate a quadratic module λ_{∗}(E) over L to it, insofar this is possible in a meaningful way.

In order to do this, we limit ourselves to the case where E is nondegenerate (see §6, Definition 3, but with K instead of 𝔬). This is not a real limitation, however. Since K is a field, every quadratic module E = (E, q) of finite dimension over K has the form E ≅ F ⊥ [0, …, 0] = F ⊥ s × [0] with F nondegenerate (see §6). So, when we have associated a specialization λ_{∗}(F) to F in a satisfactory way, it is natural to put λ_{∗}(E) = λ_{∗}(F) ⊥ s × [0]. We stay with nondegenerate quadratic K-modules, however, and now call them quadratic spaces over K, as agreed in §6, Definition 9.

Definition 1. Let E be a quadratic space over K. We say that E has good reduction with respect to λ when E ≅ K ⊗_{𝔬} M, where M is a \underline{nondegenerate} quadratic 𝔬-module.

This is the “good case”. We would like to put λ_{∗}(E): = L ⊗_{λ} M, where the tensor product is formed by means of the homomorphism λ|𝔬: 𝔬 → M, see §5, Definition 4. This tensor product can more easily be described as follows: First we go from the quadratic 𝔬-module M = (M, q) to the quadratic space M/𝔪M = (M/𝔪M, \overline{q}) over k = 𝔬/𝔪, where \overline{q} is obtained from q: M → 𝔬 in the obvious way. Next we extend scalars by means of the field extension \overline{λ}: k ↪ L, determined by λ, thus obtaining λ_{∗}(E) = L ⊗_{k} (M/𝔪M).

Is this meaningful? With a similar argument as towards the end of §3 (second proof of Theorem 2), we now show that the answer is affirmative when E is strictly regular (which means that E^{⊥} = 0, cf. §6, Definition 3).

Theorem 1. Let E be strictly regular and let M, M′ be nondegenerate quadratic 𝔬-modules with E ≅ K ⊗_{𝔬} M ≅ K ⊗_{𝔬} M′. Then M ≅ M′, and hence M/𝔪M ≅ M′/𝔪M′.

Proof. Clearly M^{⊥} = 0 and (M′)^{⊥} = 0. Hence M and M′ are also strictly regular. By the theorem at the end of §5, the space K ⊗_{𝔬} [M′ ⊥ (−M)] ≅ E ⊥ (−E) is hyperbolic. So M′ ⊥ (−M) is itself hyperbolic, by §6, Theorem 4. M ⊥ (−M) is also hyperbolic, again by the theorem at the end of §5. The modules M′ ⊥ (−M) and M ⊥ (−M) have the same rank. Hence M′ ⊥ (−M) ≅ M ⊥ (−M), and so M′ ≅ M by the Cancellation Theorem (§6, Theorem 2).

The problem remains to see that λ_{∗}(E) does not depend on the choice of the module M, when E has good reduction with respect to λ, but only is nondegenerate. Furthermore, in case of bad reduction (i.e. not good reduction), we would nevertheless like to associate a Witt class λ_{W} ({E}) over L to E in a meaningful way, just as for bilinear forms in §3 (“weak specialization”). Neither problem can be dealt with as in §3, in the first place because the Witt classes do not constitute an additive group this time. Thus we have to seek out a new path.

Lemma 1. Let 𝔬 be quadratically henselian (see §6, Definition 12) and let V = (V, q) be an anisotropic quadratic module over K.

(a) The sets

μ(V): = {x ∈ V | q(x) ∈ 𝔬} and μ_{+}(V): = {x ∈ V | q(x) ∈ 𝔪}

are 𝔬-submodules of V.

(b) For any x ∈ μ(V) and y ∈ μ_{+}(V), we have q(x + y) − q(x) ∈ 𝔪 and B_{q}(x, y) ∈ 𝔪.

Proof. (a) Let x ∈ μ(V) (resp. x ∈ μ_{+}(V)) and λ ∈ 𝔬, then clearly λx ∈ μ(V) (resp. μ_{+}(V)). So we only have to show that for any x, y ∈ μ(V) (resp. μ_{+}(V)), we have x + y ∈ μ(V) (resp. μ_{+}(V)). Let B: = B_{q}.

Let x ∈ μ(V), y ∈ μ(V). Suppose for the sake of contradiction that x + y ∉ μ(V), i.e. q(x + y) = q(x) + q(y) + B(x, y) ∉ 𝔬. Since q(x) ∈ 𝔬, q(y) ∈ 𝔬, we must have that B(x, y) ∉ 𝔬. Hence λ: = B(x, y)^{−1} ∈ 𝔪. The vectors x, λy give rise to the value matrix [{q(x) \atop 1} {1 \atop λ^{2}q(y)}]. According to §6, Lemma 6, the space Kx + Ky = Kx + Kλy is isotropic. Contradiction, since V is anisotropic. Therefore x + y ∈ μ(V).

Next, let x ∈ μ_{+}(V), y ∈ μ_{+}(V). Suppose again for the sake of contradiction that x + y ∉ μ_{+}(V). We just showed that q(x + y) ∈ 𝔬. By our assumption q(x + y) ∉ 𝔪, so that q(x + y) ∈ 𝔬^{∗}. Since q(x) ∈ 𝔪, q(y) ∈ 𝔪, we have B(x, y) ∈ 𝔬^{∗}. Write B(x, y) = λ^{−1} with λ ∈ 𝔬^{∗}. Again, the vectors x, λy give rise to the value matrix [{q(x) \atop 1} {1 \atop λ^{2}q(y)}]. Just as before, the space Kx+Ky is isotropic, according to §6, Lemma 6. Contradiction! We conclude that x + y ∈ μ_{+}(V).

(b) Suppose for the sake of contradiction that there exist vectors x ∈ μ(V), y ∈ μ_{+}(V) with q(x+y)−q(x) ∉ 𝔪. We showed above that q(x+y) ∈ 𝔬. Hence q(x + y) − q(x) = q(y) + B(x, y) ∈ 𝔬. By assumption, this element doesn’t live in 𝔪 and is thus a unit. Since q(y) ∈ 𝔪, we also have B(x, y) ∈ 𝔬^{∗}. As before, we write λ = B(x, y)^{−1} and observe that Kx + Ky is isotropic, giving a contradiction. Therefore q(x+y)−q(x) ∈ 𝔪 and B(x, y) = q(x+y)−q(x)− q(y) ∈ 𝔪 as well.

We remain in the situation of Lemma 1. For x ∈ μ(V), λ ∈ 𝔪, we have λx ∈ μ_{+}(V). Therefore

ρ(V): = μ(V)/μ_{+}(V)

is a k-vector space in a natural sense (k = 𝔬/𝔪). We define a function \overline{q}: ρ(V) → k as follows:

\overline{q}(\overline{x}): = \overline{q(x)} (x ∈ μ(V)),

where \overline{x} denotes the image of x ∈ μ(V) in ρ(V) and \overline{a} denotes the image of a ∈ 𝔬 in k. Lemma 1 tells us that the map \overline{q} is well-defined.

For x ∈ μ(V), a ∈ 𝔬, we have \overline{q}(\overline{a} \overline{x}) = \overline{q}(\overline{ax}) = \overline{q(ax)} = \overline{a}^{2}q\overline{(x)} = \overline{a}^{2}\overline{q}(\overline{x}). According to Lemma 1, the bilinear form B: = B_{q} has values in 𝔬 on μ(V)×μ(V) and values in 𝔪 on μ(V)×μ_{+}(V). Therefore, it induces on ρ(V) a symmetric bilinear form \overline{B} over k with \overline{B}(\overline{x}, \overline{y}) = \overline{B(x, y)} for x, y ∈ μ(V). A very simple calculation now shows that

\overline{q}(\overline{x} + \overline{y}) − \overline{q}(\overline{x}) − \overline{q}(\overline{y}) = \overline{B}(\overline{x}, \overline{y})

for x, y ∈ μ(V). This furnishes the proof that \overline{q} is a quadratic form on the kvector space ρ(V) with B\overline{q} = \overline{B}. The quadratic k-module (ρ(V), \overline{q}) is clearly anisotropic.

Definition 2. (𝔬 quadratically henselian.) We call the quadratic k-module ρ(V) = (ρ(V), \overline{q}) the reduction of the anisotropic quadratic K-module V with respect to the valuation ring 𝔬.

In order to associate to a quadratic space E over K, by means of the place λ: K → L ∪ ∞, a Witt class λ_{W} {E} over L, the following path presents itself now: Let 𝔬^{h} be the henselization of the valuation ring 𝔬 = 𝔬_{λ}, 𝔪^{h} the maximal ideal of 𝔬^{h} and K^{h} the quotient field of 𝔬^{h}. The residue class field 𝔬^{h}/𝔪^{h} is canonically isomorphic to k = 𝔬/𝔪 and will be identified with k. Let

λ_{W} \{E\} = \{L ⊗\overline{λ} ρ(Ker (K^{h} ⊗ E))\},

where, as before, \overline{λ}: k ↪ L is the field embedding, determined by λ. The space over L on the righthand side can then be considered to be a “weak specialization” of E with respect to λ.

All good and well, if only we knew whether the vector space

ρ(Ker (K^{h} ⊗ E))

has finite dimension! To guarantee this, we have to confine the class of allowed quadratic modules E.

In the following, v: K → Γ ∪ ∞ is a surjective valuation, associated to the valuation ring 𝔬, thus with Γ the value group of v, Γ ≅ K^{∗}/𝔬^{∗}. We use additive notation for Γ (so v(xy) = v(x) + v(y) for x, y ∈ K).

Already in §3, we agreed to view the square class group Q(𝔬) as a subgroup of Q(K), and also to regard the elements of Q(K) as one-dimensional bilinear spaces^{10} over K.

Now we choose a complement Σ of Q(𝔬) in Q(K), in other words, a subgroup Σ of Q(K) with Q(K) = Q(𝔬) × Σ. This is possible, since Q(K) is elementary abelian of exponent 2, i.e. a vector space over the field with 2 elements. Further, we choose, for every square class σ ∈ Σ, an element s ∈ 𝔬 with σ = ⟨s⟩. For σ = 1 we choose the representative s = 1. Let S be the set of all elements s. For every a ∈ K^{∗} there exists exactly one s ∈ S and elements ε ∈ 𝔬^{∗}, b ∈ K^{∗} with a = sεb^{2}. Since K^{∗}/𝔬^{∗} ≅ Γ, it is clear that S (resp. Σ) is a system of representatives of Γ/2Γ in K^{∗} (resp. Q(K)) for the homomorphism from K^{∗} to Γ/2Γ (resp. from Q(K) to Γ/2Γ), determined by v: K^{∗} → Γ.

Definition 3. A quadratic space E over K is called obedient with respect to λ (or obedient with respect to 𝔬) if there exists a decomposition

(∗)  E ≅ ⊥_{s∈S} ⟨s⟩ ⊗ (K ⊗_{𝔬} M_{s}),

where M_{s} is a nondegenerate quadratic 𝔬-module. (Of course M_{s} ≠ 0 for finitely many s only.)

Clearly this property does not depend on the choice of system of representatives S.

Remarks.

(1) Obedience is much weaker than the property “good reduction” (Definition 1).

(2) Let E and F be quadratic spaces over K, obedient with respect to 𝔬. If at least one of them is strictly regular, then E ⊥ F is obedient.

(3) If char K ≠ 2, then every quadratic space over K is strictly regular. Hence the orthogonal sum of two obedient quadratic spaces over K is again obedient.

(4) If 2 ∈ 𝔬^{∗}, i.e. char L ≠ 2, then every quadratic space E over K is obedient with respect to λ, since K also has characteristic ≠ 2 in this case. E has a decomposition in one-dimensional spaces, which are clearly obedient.

(5) If char K = 0, but char L = 2, then every quadratic space of odd dimension over K is disobedient (= not obedient) with respect to λ. To see this, let E be an obedient quadratic space with orthogonal decomposition (∗), as in Definition 3. Then the space K ⊗_{𝔬} M_{i} has quasilinear part (K ⊗_{𝔬} M_{i})^{⊥} = 0 for every i ∈ {1, …, r}. Therefore M_{i} also has quasilinear part M_{i}^{⊥} = 0, and is thus strictly regular. Hence the space M_{i}/𝔪M_{i} over k = 𝔬/𝔪 is strictly regular, and so must have even dimension. We conclude that E must have even dimension.

10Strictly speaking, square classes are isomorphism classes of one-dimensional spaces.

Let us now give an example of a two-dimensional disobedient space over a field K of characteristic 2.

First, recall the definition of the Arf-invariant Arf(φ) of a strictly regular form φ over a field k of characteristic 2. For x ∈ k, let ℘(x) := x^{2}+x. Further, let ℘(k) denote the set of all ℘(x), x ∈ k. This is a subgroup of k^{+}, i.e. k regarded as an additive group. We choose a decomposition

(∗)  φ ≅ [{a_{1} \atop 1} {1 \atop b_{1}} ] ⊥ … ⊥ [{a_{m} \atop 1} {1 \atop b_{m}} ]

and set

Arf(φ) := a_{1}b_{1} + ⋯ + a_{m}b_{m} + ℘(k) ∈ k^{+}/℘(k).

It is well-known that Arf(φ) is independent of the choice of the decomposition (∗) ([A], [S, Chap.IX. §4]).

Example. Let k be a field of characteristic 2 and K = k(t) the rational function field in one variable t over k. Further, let 𝔬 be the discrete valuation ring of K with respect to the prime polynomial t in k[t], i.e. 𝔬 = k[t]_{(t)}.

Claim. The quadratic space [{1 \atop 1} {1 \atop t^{−1}} ] over K is disobedient with respect to 𝔬.

Proof. Suppose for the sake of contradiction that this space is obedient. Then there exist a strictly regular space M over 𝔬 and an element u ∈ K^{∗} with [{1 \atop 1} {1 \atop t^{−1}} ] ≅ ⟨u⟩ ⊗ (K ⊗_{𝔬} M). By §6, Theorem 1, we have M ≅ [{α \atop 1} {1 \atop β} ] with α, β ∈ 𝔬. Hence, by §6, Lemma 2, (3), the following holds over K:

(1)  [{1 \atop 1} {1 \atop t^{−1}} ] ≅ ⟨αu⟩ [{1 \atop 1} {1 \atop αβ} ] .

Comparing Arf-invariants of both sides, shows that there exists an element a ∈ K with

(2)  t^{−1} = αβ + a^{2} + a.

Hence there exists an element a ∈ K with t^{−1} + a^{2} + a ∈ 𝔬.

If we now move to the formal power series field k((t)) ⊃ K, we easily see that such an element a does not exist: For if a = ∑_{r≥d} c_{r}t^{r} with d ∈ ℤ and

coefficients c_{r} ∈ k, c_{d} ≠ 0, then

∑_{r≥d}c_{r}^{2}t^{2r} + ∑_{r≥d}c_{r}t^{r} + t^{−1} ∈ k[[t]],

and so d < 0. The term of lowest degree, in the first sum on the left, is c_{d}^{2}t^{2d}. It cannot be compensated by other summands on the left. Therefore t^{−1} + a^{2} + a ∉ 𝔬, and the space [{1 \atop 1} {1 \atop t^{−1}} ] is disobedient.

In this proof we used the Arf-invariant. We remark that we can use easier aids: Anisotropic quadratic forms like [{1 \atop 1} {1 \atop c}] are norm forms of separable quadratic field extensions, and are as such “multiplicative”. Therefore, we can deduce immediately from (1) that [{1 \atop 1} {1 \atop t^{−1}} ] ≅ [{1 \atop 1} {1 \atop αβ} ] and then, that there exists a relation (2).

Remark. Hitherto the quasilinear parts of free quadratic modules played a predominantly negative role. This will continue to be the case. Rather often they cause a lot of complications, compared to the theory of nondegenerate bilinear modules. But sometimes quasilinear quadratic modules can do good things. Suppose again that K = k(t), 𝔬 = k[t]_{(t)} as in the example above. The space [{1 \atop 1} {1 \atop t^{−1}} ] is disobedient with respect to 𝔬. However, [{1 \atop 1} {1 \atop t^{−1}} ] ⊥ [t^{−1}] is obedient, since this space over K is isometric to [{1 \atop 1} {1 \atop 0}] ⊥ [t^{−1}], and hence to [{0 \atop 1} {1 \atop 0}] ⊥ [t].

In connection with the definition of obedience (Definition 3), we agree upon some more terminology. If E = (E, q) is a quadratic space, obedient with respect to 𝔬, then we have an (internal) orthogonal decomposition

E = ⊥_{s∈S} E_{s}

of the following kind: Every vector space E_{s} contains a free 𝔬-submodule M_{s} with E_{s} = KM_{s}, such that q|E_{s} = sq_{s} for a quadratic form q_{s}: E_{s} → K, which takes values in 𝔬 on M_{s} and for which (M_{s}, q_{s}|M_{s}) is a nondegenerate quadratic module. (Of course E_{s} ≠ 0 for finitely many s ∈ S only.)

Definition 4. We call such a decomposition E = ⊥_{s∈S} E_{s}, together with a

choice of modules M_{s} ⊂ E_{s}, a λ-modular, or also, an 𝔬-modular representation of the obedient quadratic space E.

Lemma 2. Suppose again that 𝔬 is quadratically henselian and E = (E, q) is a quadratic space over K, obedient with respect to 𝔬. Let E = ⊥_{s∈S} E_{s},

E_{s} = KM_{s} be an 𝔬-modular representation of E. Suppose furthermore that E is anisotropic. Then we have, with the notation of Lemma 1,^{11}

\begin{matrix}μ(E) = M_{1} & ⊥ ⊥_{s≠1} μ_{+}(E_{s}), \\ μ_{+}(E) = 𝔪M_{1} & ⊥ ⊥_{s≠1} μ_{+}(E_{s}). \end{matrix}

Proof. As above, we set q|E_{s} = sq_{s}. Let x be a vector of E with x ≠ 0. Then there exist finitely many pairwise distinct elements s_{1}, …, s_{r} in S, as well as primitive vectors x_{i} ∈ M_{i}: = M_{s_{i}} and scalars a_{i} ∈ K^{∗} (1 ≤ i ≤ r), such

that x = ∑_{i=1}^{r} a_{i}x_{i}. We have

11In accordance with our earlier agreement, every E_{s} is considered as a quadratic subspace of E, E_{s} = (E_{s}, q|E_{s}).

q(x) = ∑_{i=1}^{r} a_{i}^{2}s_{i}q_{i}(x_{i}),

where we used the abbreviation q_{i}: = q_{s_{i}}. According to §6, Theorem 5, we have q_{i}(x_{i}) ∈ 𝔬^{∗} for every i ∈ {1, …, r}. Hence, for i ≠ j, the values v(a_{i}^{2}s_{i}q_{i}(x_{i})) = v(a_{i}^{2}s_{i}) and v(a_{j}^{2}s_{j}q_{j}(x_{j})) = v(a_{j}^{2}s_{j}) are different. Let v(a_{k}^{2}s_{k}) be the smallest value among the v(a_{i}^{2}s_{i}), 1 ≤ i ≤ r. Then v(q(x)) = v(a_{k}^{2}s_{k}) = v(q(a_{k}x_{k})). This shows that

μ(E) = ⊥_{s∈S} μ(E_{s}), μ_{+}(E) = ⊥_{s∈S} μ_{+}(E_{s}).

For s ≠ 1, a ∈ K^{∗} and primitive y ∈ E_{s}, we have q(ay) = sa^{2}q_{s}(y) ∉ 𝔬^{∗}. Therefore, μ(E_{s}) = μ_{+}(E_{s}) for every s ∈ S \ {1}. We still have to determine the modules μ(E_{1}) and μ_{+}(E_{1}). So, let x ∈ E_{1}, x ≠ 0. We write x = ay with a ∈ K^{∗} and primitive y ∈ M_{1}. Then q(x) = a^{2}q(y) and q(y) ∈ 𝔬^{∗}. Hence q(x) ∈ 𝔬 exactly when a ∈ 𝔬, and q(x) ∈ 𝔪 exactly when a ∈ 𝔪. Therefore we conclude that μ(E_{1}) = M_{1} and μ_{+}(E_{1}) = 𝔪M_{1}.

An immediate consequence of the lemma is

Theorem 2. Let 𝔬 be quadratically henselian, and let E = (E, q) be an anisotropic quadratic space over K, obedient with respect to 𝔬. Let E = ⊥_{s∈S} E_{s}, E_{s} = KM_{s} be an 𝔬-modular representation of E. Then we have

for the reduction ρ(E) = (ρ(E), \overline{q}) of E with respect to 𝔬:

(ρ(E), \overline{q}) ≅ (M_{1}/𝔪M_{1}, \overline{q}_{1}),

where \overline{q}_{1}: M_{1}/𝔪M_{1} → k is the quadratic form over k, determined by q|M_{1}: M_{1} → 𝔬 in the obvious way.

Furthermore, the lemma tells us that M_{1} = μ(E_{1}) and 𝔪M_{1} = μ_{+}(E_{1}). In particular we have

Theorem 3. Let 𝔬 be quadratically henselian and let E = (E, q) be an anisotropic quadratic space over K, having good reduction with respect to 𝔬. Then μ(E) = (μ(E), q|μ(E)) is a nondegenerate quadratic 𝔬-module with E = K ⊗_{𝔬} μ(E), and μ_{+}(E) = 𝔪μ(E). Note that μ(E) is the only nondegenerate quadratic 𝔬-module M in E, with E = KM.

We tone down Definition 4 as follows:

Definition 5. Let E = (E, q) be a quadratic space over K, obedient with respect to λ. A λ-modular (or 𝔬-modular) decomposition of E is an orthogonal decomposition E = ⊥_{s∈S} E_{s}, in which every space ⟨s⟩ ⊗ E_{s} = (E_{s}, s · (q|E_{s}))

has good reduction with respect to λ.

If 𝔬 is quadratically henselian and E is anisotropic then, according to Theorem 3, every λ-modular decomposition corresponds to exactly one λmodular representation of E.

Lemma 3. Let 𝔬 be quadratically henselian. Let s_{1}, …, s_{r} be different elements of S and M_{1}, …, M_{r} anisotropic nondegenerate quadratic modules over 𝔬. Then

E: = ⊥_{i=1}^{r}⟨s_{i}⟩ ⊗ (K ⊗_{𝔬} M_{i})

is an anisotropic quadratic space over K.

Proof. Let a primitive vector x_{i} ∈ M_{i} be given for every i ∈ {1, …, r}. Also, let a_{1}, …, a_{r} ∈ K be scalars with

(†)  ∑_{i=1}^{r} s_{i}a_{i}^{2}q_{i}(x_{i}) = 0,

where q_{i} denotes the quadratic form on M_{i}. We must show that all a_{i} = 0.

Suppose for the sake of contradiction that this is not so. After renumbering the M_{i}, we can assume without loss of generality that for a certain t ∈ {1, …, r}, a_{i} ≠ 0 for 1 ≤ i ≤ t and a_{i} = 0 for t < i ≤ r. By §6, Theorem 5, q(x_{i}) ∈ 𝔬^{∗} for every i ∈ {1, …, t}. The values v(s_{i}a_{i}^{2}q_{i}(x)) = v(s_{i}a_{i}^{2}) with 1 ≤ i ≤ t are pairwise different. Since this contradicts equation (†), all a_{i} = 0.

We arrive at the main theorem of this chapter.

Theorem 4. Let E be a quadratic space over K, obedient with respect to 𝔬. Let

E = ⊥_{s∈S} E_{s} = ⊥_{s∈S} F_{s}

be two 𝔬-modular decompositions of E, and also let M_{1}, N_{1} be nondegenerate quadratic modules with E_{1} ≅ K ⊗_{𝔬} M_{1}, F_{1} ≅ K ⊗_{𝔬} N_{1}. Then the quadratic spaces M_{1}/𝔪M_{1} and N_{1}/𝔪N_{1} over k = 𝔬/𝔪 are Witt equivalent.

Proof. (a) For every s ∈ S\{1} we choose nondegenerate quadratic 𝔬-modules M_{s}, N_{s} with E_{s} ≅ ⟨s⟩ ⊗ (K ⊗_{𝔬} M_{s}), F_{s} ≅ ⟨s⟩ ⊗ (K ⊗_{𝔬} N_{s}). Suppose first that 𝔬 is quadratically henselian. In accordance with §6, Theorem 3, we choose Witt decompositions

M_{s} = M_{s}^{∘} ⊥ i_{s} × [{0 \atop 1} {1 \atop 0}], N_{s} = N_{s}^{∘} ⊥ j_{s} × [{0 \atop 1} {1 \atop 0}],

with M_{s}^{∘}, N_{s}^{∘} anisotropic nondegenerate quadratic 𝔬-modules. By Lemma 3 above, the nondegenerate quadratic K-modules

U: = ⊥_{s∈S}⟨s⟩ ⊗ (K ⊗_{𝔬} M_{s}^{∘}), V: = ⊥_{s∈S}⟨s⟩ ⊗ (K ⊗_{𝔬} N_{s}^{∘})

are anisotropic. Now,

E ≅ U ⊥ ( ∑_{s∈S}i_{s}) × [{0 \atop 1} {1 \atop 0}] ≅ V ⊥ ( ∑_{s∈S}j_{s}) × [{0 \atop 1} {1 \atop 0}].

Hence U ≅ V. According to Theorem 2, we have ρ(U) ≅ M_{1}^{∘}/𝔪M_{1}^{∘}, ρ(V) ≅ N_{1}^{∘}/𝔪N_{1}^{∘}. Therefore, M_{1}^{∘}/𝔪M_{1}^{∘} ≅ N_{1}^{∘}/𝔪N_{1}^{∘}. Furthermore,

M_{1}/𝔪M_{1} ≅ M_{1}^{∘}/𝔪M_{1}^{∘} ⊥ i_{1} × [{0 \atop 1} {1 \atop 0}], N_{1}/𝔪N_{1} ≅ N_{1}^{∘}/𝔪N_{1}^{∘} ⊥ j_{1} × [{0 \atop 1} {1 \atop 0}].

We conclude that M_{1}/𝔪M_{1} ∼ N_{1}/𝔪N_{1}.

(b) Suppose next that 𝔬 is arbitrary. We go from (K, 𝔬) to the henselization (K^{h}, 𝔬^{h}). (By definition, K^{h} is the quotient field of the henselian valuation ring 𝔬^{h}.) As is well-known, the valuation v: K → Γ ∪ ∞ extends uniquely to a valuation v^{h}: K^{h} → Γ ∪ ∞, which therefore again has Γ as value group. Also, 𝔬^{h} is the valuation ring of v^{h}. Consequently, S is also a system of representatives of Q(K^{h})/Q(𝔬^{h}) ≅ Γ/2Γ.

Given a free quadratic module M over 𝔬, we set – as before – M^{h}: = 𝔬^{h} ⊗_{𝔬} M and for a free quadratic module U over K, we set U^{h}: = K^{h} ⊗_{K} U. By §6, Lemma 7, the quadratic 𝔬^{h}-modules M_{s}^{h}, N_{s}^{h} are nondegenerate. Fur-

thermore, E^{h} = ⊥_{s∈S} E_{s}^{h} = ⊥_{s∈S} F_{s}^{h} and E_{s}^{h} ≅ K^{h}⊗_{𝔬^{h}} M_{s}^{h}, F_{s}^{h} ≅ K^{h}⊗_{𝔬^{h}} N_{s}^{h}

for every s ∈ S.

By above, the quadratic spaces M_{1}^{h}/𝔪^{h}M_{1}^{h} and N_{1}^{h}/𝔪^{h}N_{1}^{h} over k are equivalent. However, these spaces can be canonically identified with M_{1}/𝔪M_{1} and N_{1}/𝔪N_{1} (cf. the end of the proof of §6, Lemma 7). We conclude that M_{1}/𝔪M_{1} ∼ N_{1}/𝔪N_{1}.

Definition 6. Let E be a quadratic space over K, obedient with respect to λ. If E = ⊥_{s∈S} E_{s} is a λ-modular decomposition of E and M_{1} a nondegen-

erate quadratic 𝔬-module with E_{1} ≅ K ⊗_{𝔬} M_{1}, then we call the quadratic space L ⊗_{λ} M_{1} a weak specialization of E with respect to λ. (As before, ⊗_{λ} denotes a base extension with respect to the homomorphism λ|𝔬: 𝔬 → L.) By Theorem 4, the space L ⊗_{λ} M_{1} is uniquely determined by E and λ, up to Witt equivalence. We denote its Witt class by λ_{W} (E), i.e.

λ_{W} (E): = {L ⊗_{λ} M_{1}}.

(“W” as in “Witt” or “weak”.)

If char K = 2, then M_{1}/𝔪M_{1} is a quadratic space over k, to be sure. Nevertheless L ⊗_{λ} M_{1} = L ⊗\overline{λ} (M_{1}/𝔪M_{1}) can be degenerate. In this case, we only have λ_{W} (E) ∈ Ŵq(L) (see §6, from Definition 10 onwards). If char K ≠ 2, this cannot happen since M_{1}/𝔪M_{1} is strictly regular in this case. So now, λ_{W} (E) ∈ Wq(L).

Theorem 5. If E and E′ are quadratic spaces over K, obedient with respect to λ, and if E ∼ E′, then λ_{W} (E) = λ_{W} (E′).

Proof. Suppose without loss of generality that dim E ≤ dim E′. Then E′ ≅ E ⊥ r × [{0 \atop 1} {1 \atop 0}] for a certain r ∈ ℕ_{0}. If we choose a nondegenerate 𝔬-module M_{1} for E, as in Definition 6, then M′_{1}: = M_{1} ⊥ r × [{0 \atop 1} {1 \atop 0}] is a possibe choice for E′. (Here we consider [{0 \atop 1} {1 \atop 0}] over 𝔬 instead of over K.) Therefore, L ⊗_{λ} M′_{1} ∼ L ⊗_{λ} M_{1}.

Example. Let K = k(t_{1}, …, t_{n}) be the rational function field in n variables t_{1}, …, t_{n} over an arbitrary field k. For every multi-index α = (α_{1}, …, α_{n}) ∈ ℕ_{0}^{n}, let t^{α} denote the monomial t_{1}^{α_{1}} … t_{n}^{α_{n}}. We order the abelian group ℤ^{n} lexicographically and then have exactly one valuation v: K → ℤ^{n} ∪ ∞ with v(t^{α}) = α for every α ∈ ℕ_{0}^{n}. Its valuation ring contains the field k, and the residue class field 𝔬/𝔪 coincides with k. Let λ: K → k ∪ ∞ be the canonical place corresponding to 𝔬. Furthermore, let A be the set of all multi-indices (α_{1}, …, α_{n}) with α_{i} ∈ {0, 1} for every i, in other words A = {0, 1}^{n} ⊂ ℕ_{0}^{n}. As system of representatives S, in the above sense, we take the set of t^{α} with α ∈ A.

Suppose now that we are given a family (F_{α} | α ∈ A) = F of 2^{n} quadratic spaces over k. We construct the space

E: = ⊥_{α∈A} ⟨t^{α}⟩ ⊗ (K ⊗_{k} F_{α})

over K. For every α ∈ A, ⟨t^{α}⟩ ⊗ E is obedient with respect to λ and

λ_{W} (⟨t^{α}⟩ ⊗ E) = {F_{α}}.

Hence the family F can be recovered from the space E, up to Witt equivalence. If all F_{α} are anisotropic, we even get the F_{α} back from E as kernel spaces of the Witt classes λ_{W} (⟨t^{α}⟩⊗E). In this case, E is anisotropic as well. This can easily be seen, using an argument similar to the one used in the proof of Lemma 3. One could say that the family F is “stored” in the space E over k((t_{1}, …, t_{n})).

This simple example reminisces of Springer’s Theorem in §1. Without our theory of weak specialization, we can use Springer’s Theorem and induction on n to show that the F_{α} are uniquely determined by E (at least when char k ≠ 2). This is so, because the valuation v is “n-fold discrete”. For more complicated value groups, we cannot fall back on Springer’s Theorem for specialization arguments.

We return to the situation of arbitrary places λ: K → k ∪ ∞.

Definition 7. (a) We denote the sets of Witt classes {E} of obedient quadratic spaces (with respect to 𝔬) and obedient strictly regular quadratic spaces (with respect to 𝔬) by W̃q(K, 𝔬) and Wq(K, 𝔬) respectively. These are thus subsets of the sets W̃q(K) and Wq(K), introduced in §6, Definition 7b.

(b) We define a map λ_{W}: W̃q(K, 𝔬) → Ŵq(L) by setting λ_{W} ({E}): = λ_{W} (E). This map is well-defined by Theorem 5.

Clearly Wq(K, 𝔬) is a subgroup of the abelian group Wq(K). The group Wq(K, 𝔬) acts on the set W̃q(K, 𝔬) by restriction of the action of Wq(K) on W̃q(K), explained in §6. If char K ≠ 2, then, of course, W̃q(K, 𝔬) = Wq(K, 𝔬), and λ_{W} maps Wq(K, 𝔬) to Wq(L).

Remark. Let E and F be quadratic spaces over K, obedient with respect to λ and suppose that E is strictly regular. Obviously we then have

λ_{W} (E ⊥ F) = λ_{W} (E) + λ_{W} (F).

Therefore, restricting the map λ_{W}: W̃q(K, 𝔬) → Ŵq(L) gives rise to a homomorphism from Wq(K, 𝔬) to Wq(L), which we also denote by λ_{W}. The map λ_{W}: W̃q(K, 𝔬) → Ŵq(L) is equivariant with respect to the homomorphism λ_{W}: Wq(K, 𝔬) → Wq(L).

In §3, we got a homomorphism λ_{W}: W (K) → W (L), using a different method. Given a bilinear space E = (E, B) over K, does there exist a description of the Witt class λ_{W} ({E}), analogous to our current Definition 5?

First of all, when 2 ∈ 𝔬^{∗}, i.e. char L ≠ 2, we find complete harmony between §3 and §6. In this case, bilinear spaces over K and L are the same objects as quadratic spaces, and every such space over K is obedient with respect to λ.

Theorem 6. If char L ≠ 2, the homomorphism λ_{W}: W (K) → W (L) from §3 coincides with the homomorphism λ_{W}: Wq(K) → Wq(L), defined just now.

Proof. It suffices to show that for a one-dimensional space [a] = ⟨2a⟩, the element λ_{W} ({⟨2a⟩}) from §3 coincides with the currently defined Witt class λ_{W} ([a]). If a ∈ 𝔬^{∗}, then λ_{W} ({⟨2a⟩}) = {⟨2λ(a)⟩} = {[λ(a)]}, according to §3. However, if ⟨a⟩ ∉ Q(𝔬), then ⟨2a⟩ ∉ Q(𝔬) and λ_{W} ({⟨2a⟩}) = 0. We obtain the same values for λ_{W} ([a]), using Definition 6 above.

And so, when char L ≠ 2, the work performed hitherto gives us a new – more conceptual – proof of §3, Theorem 1.

What is the situation when char L = 2? First, we encounter in all generality definitions for bilinear spaces, analogous to Definitions 1 and 5.

Definition 8. Let E = (E, B) be a bilinear space over K.

(a) E has good reduction with respect to λ (or with respect to 𝔬) when E ≅ K ⊗_{𝔬} M for a bilinear space M over 𝔬. {Note: this is just a translation of the original definition of “good reduction” of §1 (= §3, Definition 1) in geometric language.}

(b) A λ-modular (or 𝔬-modular) decomposition of E is an orthogonal decomposition E = ⊥_{s∈S} E_{s}, in which every space ⟨s⟩ ⊗ E_{s} has good reduction

with respect to λ.

In contrast to the quadratic case, every bilinear space E over K has λmodular decompositions. This is obvious, since E is the orthogonal sum of one-dimensional bilinear spaces and copies of the space ({0 \atop 1} {1 \atop 0}), see §2.

Let λ_{W}: W (K) → W (L) be the map, introduced in §3. When E is a bilinear space over K, we set λ_{W} (E): = λ_{W} ({E}) ∈ W (L).

Theorem 7. Let E be a bilinear space over K and let E = ⊥_{s∈S} E_{s} be a

λ-modular decomposition of E. Also, let M_{1} be a bilinear space over 𝔬 with E_{1} ≅ K ⊗_{𝔬} M_{1}. Then

λ_{W} (E) = {L ⊗_{λ} M_{1}}.

Proof. We have λ_{W} (E) = ∑_{s∈S} λ_{W} (E_{s}). By §3, Theorem 2, we know that

λ_{W} (E_{1}) = {L ⊗_{λ} M_{1}}. It remains to show that λ_{W} (E_{s}) = 0 for every s ∈ S with s ≠ 1. Hence we have to show that if ⟨s⟩ ∈ Q(K) is a square class, not in Q(𝔬), and if N is a bilinear space over 𝔬, then λ_{W} (⟨s⟩ ⊗ (K ⊗_{𝔬} N)) = 0.

By a lemma from §3, N has an orthogonal basis when the bilinear space N/𝔪N over k is not hyperbolic. In this case we have

⟨s⟩ ⊗ (K ⊗_{𝔬} N) ≅ ⟨sε_{1}, …, sε_{r}⟩

with units ε_{i} ∈ 𝔬^{∗}. By definition of λ_{W}, we have λ_{W} (⟨sε_{i}⟩) = 0 for all i ∈ {1, …, r}, hence λ_{W} (⟨s⟩ ⊗ (K ⊗_{𝔬} N)) = 0. If N/𝔪N is hyperbolic, we construct the space N′: = N ⊥ ⟨1⟩ over 𝔬. Since N′ has an orthogonal basis, we get λ_{W} (⟨s⟩⊗(K ⊗_{𝔬} N′)) = 0. Since λ_{W} (⟨s⟩⊗⟨1⟩) = 0 as well, we conclude that λ_{W} (⟨s⟩ ⊗ (K ⊗ N)) = 0.

Theorem 7 allows us to speak of “weak specialization” of bilinear spaces, in analogy with Definition 6 for quadratic spaces.

Definition 9. Let E be a bilinear space over K. Also, let E = ⊥_{s∈S} E_{s}

be a λ-modular decomposition of E and M_{1} a bilinear space over 𝔬 with E_{1} ≅ K ⊗_{𝔬} M_{1}. Then we call the space L ⊗_{λ} M_{1} a weak specialization of the space E with respect to λ.

According to Theorem 7, a weak specialization of E with respect to λ is completely determined by E and λ up to Witt equivalence. One could ask if this can be shown in a direct – geometric – way, comparable to how we have done this for obedient quadratic spaces above. Reversing direction, this would give a new proof of §3, Theorem 1, also when char L = 2.

We leave this question open. After all, we have a proof of §3, Theorem 1, and with it a weak specialization theory for bilinear spaces. This theory is more satisfying than the corresponding theory for quadratic spaces, in that we do not have to demand obedience of the spaces.

We can now also make a statement about Problem 3b of §1. Let char L = 2 and let char K = 0. Let (E, q) be a quadratic space, obedient with respect to λ. Associated to it, we have the bilinear space (E, B_{q}). It may be that (E, q) and (E, B_{q}) are in principle the same object, but it does make a difference wether we weakly specialize E as a quadratic or a bilinear space. What is better?

Let (M, q) be a regular (= strictly regular) quadratic 𝔬-module. According to §6, Theorem 1, we have a decomposition

(M, q) ≅ [{a_{1} \atop 1} {1 \atop b_{1}} ] ⊥ … ⊥ [{a_{r} \atop 1} {1 \atop b_{r}} ].

(Note: (M/𝔪M, \overline{q}) is strictly regular, and thus has even dimension.) Then

(M, B_{q}) ≅ ({2a_{1} \atop 1} {1 \atop 2b_{1}} ) ⊥ … ⊥ ({2a_{r} \atop 1} {1 \atop 2b_{r}} ).

Hence,

L ⊗_{λ} (M, q) ≅ [{λ(a_{1}) \atop 1} {1 \atop λ(b_{1})}] ⊥ … ⊥ [{λ(a_{r}) \atop 1} {1 \atop λ(b_{r})}].

However, L ⊗_{λ} (M, B_{q}) ≅ r × ({0 \atop 1} {1 \atop 0}).

Therefore, a weak specialization of the space (E, B_{q}) is always hyperbolic and so gives hardly any information about (E, q). However, a weak specialization of (E, q) with respect to λ can give an interesting result.

Final Consideration. A last word about the central Definition 3 of an obedient quadratic space. It seems obvious to formally weaken it, by considering quadratic modules E instead of spaces E over K, subject only to the requirements that, as a K-vector space, E should have finite dimension and a decomposition (∗) as presented there. But then E is already nondegenerate. We namely have

QL(E) ≅ ⊥_{s∈S} ⟨s⟩ ⊗ (K ⊗_{𝔬} QL(M_{s})),

and as in the proof of Lemma 3, we see that QL(E) is anisotropic (without the requirement that 𝔬 is quadratically henselian). Thus these “obedient quadratic modules” are the same objects as those given by Definition 3.

60 Chapter I. Fundamentals of Specialization Theory

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§8 Good Reduction

As before, we let λ: K → L ∪ ∞ be a place, 𝔬 = 𝔬_{λ} the valuation ring of K, 𝔪 its maximal ideal, k = 𝔬/𝔪 its residue class field and v a valuation on K, associated to 𝔬, with value group Γ. Also, let Σ ⊂ Q(K) be a complement of Q(𝔬) in Q(K) and S a system of representatives of Σ in K^{∗} (with 1 ∈ S), as introduced in §7.

From now on, we call a nondegenerate quadratic 𝔬-module a quadratic space over 𝔬. In case 𝔬 = K, we already used this terminology in §7. Recall further the concept of a bilinear space over 𝔬 (§3, Definition 4).

If E is a quadratic (resp. bilinear) space over K then, according to Definitions 1 and 8 in §7, E has good reduction with respect to λ (or: with respect to 𝔬) if there exists a quadratic (resp. bilinear) space M over 𝔬 with E ≅ L ⊗_{𝔬} M.

Theorem 1. Let E be a quadratic or bilinear space over K, which has good reduction with respect to λ, and let M, M′ be quadratic (resp. bilinear) spaces over 𝔬 with E ≅ K ⊗_{𝔬} M ≅ K ⊗_{𝔬} M′. Then the k-spaces M/𝔪M and M′/𝔪M′ are isometric in the quadratic case and stably isometric^{12} in the bilinear case. Therefore we also have, L ⊗_{λ} M ≅ L ⊗_{λ} M′ resp. L ⊗_{λ} M ≈ L ⊗_{λ} M′.

Proof. By the theory of §7 (Theorems 4 and 7), M/𝔪M and M′/𝔪M′ are Witt equivalent. Since these spaces have the same dimension, namely dim E, they are isometric resp. stably isometric.

In the following, the words “good reduction” are used so often, that it is appropriate to introduce an abbreviation. From now on, we mostly write GR instead of “good reduction”.

Definition 1. Let E be a quadratic or bilinear space over K, which has GR with respect to λ, and let M be a quadratic, resp. bilinear, space over 𝔬 with E ≅ K ⊗_{𝔬} M. Then we denote the quadratic, resp. bilinear, module L⊗_{λ} M by λ_{∗}(E) and call it “the” specialization of E with respect to λ.

In the bilinear case, λ_{∗}(E) is nondegenerate, and thus a space. This is true in the quadratic case as well, as long as char K ≠ 2. If char K = 2

12See §2 for the term “stably isometric”.

however, λ_{∗}(E) can be degenerate (cf. our discussion about λ_{W} (E) in §7 after Definition 6).

This terminology is convenient, but sloppy. According to Theorem 1, λ_{∗}(E) is only determined by E up to isometry in the quadratic case, and in the bilinear case even only up to stable isometry. Anyway, in what follows, we are interested in quadratic spaces (or modules) only up to isometry, and in bilinear spaces almost always only up to stable isometry.

Note that in the bilinear case, Definition 1 is only a translation of §3, Definition 3 in geometric language.

Remark. Let F and G be bilinear spaces over K, which have GR with respect to λ. Then F ⊥ G clearly also has GR with respect to λ, and

λ_{∗}(F ⊥ G) ≈ λ_{∗}(F) ⊥ λ_{∗}(G).

For quadratic spaces, we have to realize that (when char K = 2) the orthogonal sum of two spaces over 𝔬 can possibly be degenerate and thus need not again be a space. However, if F and G are quadratic spaces, which have GR with respect to λ, and if F is strictly regular, then F ⊥ G has GR with respect to λ and

λ_{∗}(F ⊥ G) ≅ λ_{∗}(F) ⊥ λ_{∗}(G).

For further applications, the following theorem is of the utmost importance.

Theorem 2. (a) Let F and G be bilinear spaces over K. If the spaces F and F ⊥ G have GR with respect to λ, then G also has GR with respect to λ.

(b) Let F and G be quadratic spaces over K. Suppose that F is strictly regular. If F and F ⊥ G have GR with respect to λ, then G also has GR with respect to λ.

Proof. Part (a) has already been proved in §3, Theorem 4. Upon replacing every occurrence of “metabolic” by “hyperbolic”, the argument there also yields a proof of part (b). (Use Theorem 3 and Lemma 3 in §6.)

Corollary. Let E and F be bilinear (resp. quadratic) spaces with E ∼ F. If E has GR with respect to λ, then so has F, and λ_{∗}(E) ∼ λ_{∗}(F). Besides, if E ≈ F (resp. E ≅ F), then λ_{∗}(E) ≈ λ_{∗}(F) (resp. λ_{∗}(E) ≅ λ_{∗}(F)).

This was already established in §3 for the bilinear case. The quadratic case can be proved similarly.

In part (b) of Theorem 2, the assumed strict regularity of F is essential.

Example. Let char K = 2 and let E be a 3-dimensional quadratic space over K with basis e, f, g and corresponding value matrix [{1 \atop 1} {1 \atop a}] ⊥ [ε], where a ∈ 𝔬

and ε ∈ 𝔬^{∗}. Let F: = E^{⊥} = Kg and G: = Ke + K(f + cg) ≅ [{1 \atop 1} {1 \atop a+c^{2}ε}] for some c ∈ K. We have E = F ⊥ G. The quadratic spaces E and F have good reduction, but G can have bad reduction because the possibility exists that the Arf-invariant a + c^{2}ε + ℘K, ℘K: = {x^{2} + x | x ∈ K} does not contain an element of 𝔬.

For instance, let K = k(t) where k is an imperfect field of characteristic 2 and t is an indeterminate. As in the example of a disobedient space, §7, let 𝔬 = k[t]_{(t)}. We choose ε ∈ k^{∗} such that ε is not a square in k. Furthermore, we choose a = 0, c = t^{−2}. We have εt^{−2} + x^{2} + x ∉ 𝔬 for every x ∈ K. This is immediately clear when we look at the power series expansions in k((t)): Let x = ∑_{n≥d} c_{n}t^{n} for some d ∈ ℤ, all c_{n} ∈ k and c_{d} ≠ 0. If it was true that

εt^{−2} + x^{2} + x ∈ 𝔬, then we would have

εt^{−2} + ∑_{n≥d} c_{n}^{2}t^{2n} + ∑_{n≥d} c_{n}t^{n} ∈ k[[t]].

But then d < 0, and so d = −1. It would follow that ε = c_{d}^{2}, an impossibility.

Theorem 3. Let E be a quadratic space over K. E has GR with respect to λ if and only if QL(E) has GR with respect to λ and there exists a decomposition E = F ⊥ QL(E) such that F has GR with respect to λ. In this case λ_{∗}(E) ≅ λ_{∗}(F) ⊥ λ_{∗}(QL(E)).

Proof. Suppose first that we have a decomposition E = F ⊥ QL(E), in which both F and QL(E) have GR with respect to λ. Then, by the remark preceding Theorem 2, E has GR with respect to λ and λ_{∗}(E) ≅ λ_{∗}(F) ⊥ λ_{∗}(QL(E)), because F is strictly regular.

Next, suppose that E has GR. We choose a quadratic space M over 𝔬 with E ≅ K ⊗_{𝔬} M and a decomposition M = N ⊥ QL(M). From the definition of spaces over 𝔬, in other words of nondegenerate quadratic 𝔬modules (§6, Definition 3), follows immediately that QL(M) is nondegenerate and even that N is strictly regular. Now E ≅ K ⊗_{𝔬} N ⊥ K ⊗_{𝔬} QL(M), so that QL(E) ≅ K ⊗_{𝔬} QL(M). Therefore QL(E) has GR. Furthermore F: = K ⊗_{𝔬} N has GR and is strictly regular.

If F and G are bilinear spaces over K, which have GR with respect to λ, then we know from §3 that F ⊗ G also has GR with respect to λ and that

λ_{∗}(F ⊗ G) ≈ λ_{∗}(F) ⊗ λ_{∗}(G),

see §3, Theorem 6. As explained in §5, given a bilinear space F and a quadratic space G over K (or 𝔬), we can also construct a quadratic module F ⊗ G over K (resp. 𝔬), which can however be degenerate when char K = 2. For F ⊗ G to be again a space, we must require that G is strictly regular. Analogous to the above statement, we have

Theorem 4. Let F be a bilinear space over K, which has GR with respect to λ and let G be a quadratic space, having GR with respect to λ. In case char K = 2, suppose furthermore that G is strictly regular. Then F ⊗ G als has GR with respect to λ and

λ_{∗}(F ⊗ G) ≅ λ_{∗}(F) ⊗ λ_{∗}(G).

Proof. We choose spaces M and N over 𝔬 with F ≅ K ⊗_{𝔬} M, G ≅ K ⊗_{𝔬} N. Then N is strictly regular. Therefore M ⊗_{𝔬} N is a strictly regular quadratic space over 𝔬. Furthermore, F ⊗ G ≅ K ⊗_{𝔬} (M ⊗_{𝔬} N). Hence, F ⊗ G has GR and

λ_{∗}(F ⊗ G) ≅ L ⊗_{λ} (M ⊗_{𝔬} N) ≅ (L ⊗_{λ} M) ⊗_{L} (L ⊗_{λ} N) ≅ λ_{∗}(F) ⊗ λ_{∗}(G).

Remark. λ_{∗}(F) is only determined up to stable isometry and λ_{∗}(F ⊗ G) only up to isometry. The theorem is valid – as the proof shows – for every choice of λ_{∗}(F). More generally we have: if E and E′ are bilinear space over K with E ≈ E′, and if F is a strictly regular quadratic space over K, then E ⊗F ≅ E′ ⊗F. This follows from the fact that there exists a bilinear space U over K with E ⊥ U ≅ E′ ⊥ U, implying that (E ⊗ F) ⊥ (U ⊗ F) ≅ (E′ ⊗ F) ⊥ (U ⊗ F), and so E ⊗ F ≅ E′ ⊗ F by the Cancellation Theorem.

The currently developed notion of the specialization λ_{∗}(E) of a space E which has GR, allows us to complete our understanding of the results from §7 about obedience and weak specialization.

Theorem 5. Let E be a quadratic module over K. E is nondegenerate and obedient with respect to λ if and only if

(∗)  E ≅ ⊥_{s∈S} ⟨s⟩ ⊗ F_{s},

where the F_{s} are spaces over K, which have GR with respect to λ, only finitely many of them being nonzero. For every decomposition of the form (∗), we have λ_{W} (E) = {λ_{∗}(F_{1})}.

Proof. If E is a space and is obedient with respect to λ, we clearly have a decomposition as above. The equality λ_{W} (E) = {λ_{∗}(F_{1})} follows from the definition of λ_{W} (E) in §7 and the definition of λ_{∗}(F_{1}).

Now let (F_{s} | s ∈ S) be a family of spaces, having GR with respect to λ and F_{s} = 0 for almost all s ∈ S. Let E: = ⊥_{s∈S} ⟨s⟩ ⊗ F_{s}. Then, the quadratic

module E over K has finite dimension and E^{⊥} = ⊥_{s∈S} ⟨s⟩ ⊗ F_{s}^{⊥}. According to

the Final Consideration of §7, E^{⊥} is anisotropic and therefore nondegenerate. It follows immediately from §7, Definition 3 that E is obedient with respect to λ.

We call every decomposition of the form (∗), having the properties of Theorem 5, a λ-modular decomposition of E. This terminology is a bit sloppier than in §7, in the sense that we no longer discriminate between internal and external orthogonal sums. We talk about λ-modular decompositions of bilinear spaces in a similar fashion.

Theorem 6. Let F be a bilinear space over K, which has GR with respect to λ.

(i) If G is a bilinear space over K, then

(†)  λ_{W} (F ⊗ G) = λ_{W} (F)λ_{W} (G) = {λ_{∗}(F)}λ_{W} (G).

(ii) If G is a strictly regular quadratic space over K, obedient with respect to λ, then F ⊗ G is also obedient with respect to λ, and (†) holds again.

Proof. (ii) Let G ≅ ⊥_{s∈S} ⟨s⟩ ⊗ G_{s} be a λ-modular decomposition of G. Every

G_{s} has GR with respect to λ and is strictly regular. By Theorem 4, F ⊗ G_{s} has GR with respect to λ and λ_{∗}(F ⊗ G_{s}) ≅ λ_{∗}(F) ⊗ λ_{∗}(G_{s}). Hence,

F ⊗ G ≅ ⊥_{s∈S} ⟨s⟩ ⊗ (F ⊗ G_{s})

is a λ-modular decomposition of F ⊗ G and λ_{W} (F ⊗ G) = {λ_{∗}(F ⊗ G_{1})} = {λ_{∗}(F) ⊗ λ_{∗}(G_{1})} = {λ_{∗}(F)}{λ_{∗}(G_{1})} = λ_{W} (F)λ_{W} (G).

The proof of (i) is analogous. Less care is needed here than for (ii).

Similarly we can show,

Theorem 7. Let F be a bilinear space and G a strictly regular quadratic space over K. Suppose that the space G has GR with respect to λ. Then F ⊗ G is obedient with respect to λ, and

λ_{W} (F ⊗ G) = λ_{W} (F)λ_{W} (G) = λ_{W} (F){λ_{∗}(G)}.

To conclude this section, we have a look at the reduction behaviour of quadratic spaces under base field extensions. The following – almost banal – theorem, together with Theorem 2(b) above, lays the foundations upon which we will build the generic splitting theory of regular quadratic spaces in the next section.

Theorem 8. Let K′ ⊃ K be a field extension and μ: K′ → L∪∞ an extension of the place λ: K → L ∪ ∞. Let E be a regular quadratic space over K, which has GR with respect to λ. The space K′ ⊗_{K} E has GR with respect to μ and μ_{∗}(K′ ⊗_{K} E) ≅ λ_{∗}(E).

Proof. Let 𝔬′ be the valuation ring associated to μ. Furthermore, let M be a regular quadratic space over 𝔬 with E ≅ K ⊗_{𝔬} M. Then 𝔬′ ⊗_{𝔬} M is a regular quadratic space over 𝔬′ (sic!), and K′ ⊗_{K} E ≅ K′ ⊗_{𝔬} M = K′ ⊗_{𝔬′} (𝔬′ ⊗_{𝔬} M). Therefore, K′ ⊗_{K} E has GR with respect to μ and

μ_{∗}(K′ ⊗_{K} E) ≅ L ⊗_{μ} (𝔬′ ⊗_{𝔬} M) = L ⊗_{λ} M ≅ λ_{∗}(E).

If we only require that E is nondegenerate instead of regular, the statement of Theorem 8 becomes false. Looking for a counterexample, we can restrict ourselves to quasilinear spaces, by Theorems 3 and 8.

Example. Let k be an imperfect field of characteristic 2 and let a be an element of k which is not a square. Consider the power series field K = k((t)) in one indeterminate t. Let λ_{0}: K → k ∪ ∞ be the place with valuation ring 𝔬 = k[[t]], which maps every power series f(t) ∈ 𝔬 to its constant term f(0). The quasilinear quadratic 𝔬-module M = [1, a + t] is nondegenerate. For, if f(t) = ∑_{i≥0} b_{i}t^{i} and g(t) = ∑_{i≥0} c_{i}t^{i} are elements of 𝔬, whose constant terms

b_{0}, c_{0} are not both zero, then f(t)^{2} + (a + t)g(t)^{2} has nonzero constant term b_{0}^{2} + ac_{0}^{2}, and is thus a unit in 𝔬. Therefore, axiom (QM2) of §6, Definition 3 is satisfied.

Hence, the space E: = K⊗_{𝔬}M has GR with respect to λ_{0}, and (λ_{0})_{∗}(E) ≅ [1, a]. Consider now L: = k(\sqrt{a}) and the place λ: = j ∘ λ_{0}: K → L ∪ ∞, obtained by composing λ_{0} with the inclusion j: k ↪ L. Then E also has GR with respect to λ, and over L we have,

λ_{∗}(E) = L ⊗_{k} (λ_{0})_{∗}(E) ≅ [1, a] ≅ [1, 1] ≅ [1, 0].

Finally, let K′: = K(\sqrt{a}) = L((t)) and let μ: K′ → L ∪ ∞ be the place with valuation ring 𝔬′: = L[[t]], which again maps every power series f(t) ∈ 𝔬′ to its constant term f(0). μ extends the place λ. Over K′, we have

K′ ⊗_{K} E = [1, (\sqrt{a})^{2} + t] ≅ [1, t].

Therefore, K′ ⊗_{K} E is obedient with respect to μ and μ_{W} (K′ ⊗_{K} E) = {[1]}. It is now clear that K′ ⊗_{K} E has bad reduction with respect to μ, since the Witt class μ_{W} (K′ ⊗_{K} E) would contain a two-dimensional module otherwise.

Remark. This argumentation shows nicely that it is profitable to give such an elaborate definition of Witt equivalence of degenerate quadratic modules over fields, as done in §6, Definition 10.

Chapter II

Generic Splitting Theory

68 Chapter II. Generic Splitting Theory

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§9 Generic Splitting of Regular Quadratic Forms

From now on we leave the geometric arena behind, and mostly talk of quadratic and bilinear forms, instead of spaces, over fields. The importance of the geometric point of view was to bring quadratic and bilinear modules over valuation rings into the game. For our specialization theory, these modules were merely an aid however, and their rôle has now more or less ended.

We should indicate one problem though, which occurs when we make the transition from quadratic spaces to quadratic forms: If φ(x_{1}, …, x_{n}) and ψ(x_{1}, …, x_{n}) are two quadratic forms over a field K, they are isometric, ψ ≅ φ, if the polynomial ψ emerges from φ through a linear coordinate transformation. Now, if φ – i.e. the space (E, q) associated to φ – is degenerate, then it is possible that not all of the coordinates x_{1}, …, x_{n} (n: = dim E) occur in the polynomial ψ (and possibly neither in φ). The dimension of ψ in the naive sense, i.e. the number of occurring variables, can be smaller than the dimension of E. In order to recover the space (E, q) from ψ, one would have to attach a “virtual dimension” n to ψ, pretty horrible!

As soon as one allows degenerate forms, the geometric language is more precise – and thus more preferred – than the “algebraic” language. But now we only consider regular quadratic forms (ˆ= regular quadratic spaces) over fields, so that the forms are guaranteed to remain nondegenerate under base field extensions.

Many concept which we introduced for quadratic spaces over fields (§5 – §8), will now be used for quadratic forms, usually without any further comments. In what follows, a “form” is always a regular quadratic form over a field. We recall once more the concepts GR (= good reduction) and specialization of forms and try to be as down to earth as possible.

So, let φ be a form over a field K. If dim φ is even (resp. odd), then φ = [a_{ij}] (resp. φ ≅ [a_{ij}] ⊥ [c]), where (a_{ij}) is a symmetric (2m) × (2m)- matrix, such that (r: = 2m)

det ( \begin{matrix}2a_{11} & a_{12} & … & a_{1r} \\ a_{21} & 2a_{22} & … & a_{2r} \\ ⋮ & ⋮ & & ⋮ \\ a_{r1} & a_{r2} & … & 2a_{rr} \end{matrix} ) ≠ 0

and c ≠ 0. (See the beginning of §5 for the notation used here.)

Let λ: K → L∪∞ be a place and 𝔬 = 𝔬_{λ} its valuation ring. The form φ has GR with respect to λ when φ ≅ [a_{ij}], resp. φ ≅ [a_{ij}] ⊥ [c], such that all a_{ij} are in 𝔬 and c, as well as the determinant above, are units in 𝔬. We then have λ_{∗}(φ) ≅ [λ(a_{ij})] resp. λ_{∗}(φ) ≅ [λ(a_{ij})] ⊥ [λ(c)]. By §6, Theorem 1(b), [a_{ij}] is the orthogonal sum of m binary forms [{a_{i} \atop 1} {1 \atop b_{i}} ] (1 ≤ i ≤ m) with a_{i} ∈ 𝔬, b_{i} ∈ 𝔬, 1 − 4a_{i}b_{i} ∈ 𝔬^{∗}. Thus, after a coordinate transformation, we have

(∗) φ(x_{1}, …, x_{n}) = ∑_{i=1}^{m}(a_{i}x_{2i}^{2} + x_{2i}x_{2i+1} + b_{i}x_{2i+1}^{2})  ( +cx_{2m+1}^{2}).

We obtain (λ_{∗}φ)(x_{1}, …, x_{n}) from this form, upon replacing the coefficients a_{i}, b_{i}, c by λ(a_{i}), λ(b_{i}), λ(c).

Let K′ ⊃ K be a field extension. If φ = φ(x_{1}, …, x_{n}) is a form over K, we write φ ⊗ K′ or φ ⊗_{K} K′ for this form considered over K′. If (E, q) is the quadratic space associated to φ, i.e. φ ˆ= (E, q), then the basis extension (K′ ⊗_{K} E, q_{K′}) of (E, q) is the quadratic space associated to φ ⊗ K′.^{13}

Let μ: K′ → L ∪ ∞ be an extension of the place λ: K → L ∪ ∞. If φ has GR with respect to λ, then φ ⊗ K′ has GR with respect to μ and μ_{∗}(φ ⊗ K′) ≅ λ_{∗}(φ). This is clear now and was already established in §8, Theorem 8 anyway.

We can extend the observation we made, at the start of our treatment of generic splitting in §4, to regular quadratic forms. Thus let k be a field and φ a form over k. Furthermore, let K and L be fields, containing k, and let λ: K → L ∪ ∞ be a place over k. Then φ ⊗ K has GR with respect to λ and λ_{∗}(φ ⊗ K) ≅ φ ⊗ L.

We use H to denote the quadratic form [{0 \atop 1} {1 \atop 0}], regardless of the field we are working in. The bilinear form B_{φ}, associated to φ = [{0 \atop 1} {1 \atop 0}], is ({0 \atop 1} {1 \atop 0}). We denote this bilinear form henceforth by H̃. (This notation differs from §4!) If

φ ⊗ K ≅ φ_{1} ⊥ r_{1} × H

is the Witt decomposition of φ ⊗ K, then the form φ_{1} has GR with respect to λ by §8, Theorem 2(b) and we have

φ ⊗ L ≅ λ_{∗}(φ_{1}) ⊥ r_{1} × H.

Therefore, ind(φ ⊗ L) ≥ ind(φ ⊗ K). If K and L are specialization equivalent over k (see §4, Definition 1), then

ind(φ ⊗ L) = ind(φ ⊗ K) and ker(φ ⊗ L) ≅ λ_{∗}(φ_{1}).

We transfer the definition of generic zero field (§4, Definition 2) literally to the current situation. Just as in §4 we define, for n: = dim φ ≥ 2, a field extension k(φ) of k as follows: If n > 2 or n = 2 and φ ≅ H̸, let k(φ) be the quotient field of the integral domain k[X_{1}, …, X_{n}]/(φ(X_{1}, …, X_{n})). {It is easy to see that the polynomial φ(X_{1}, …, X_{n}) is irreducible.} If φ ≅ H however, i.e. φ(X_{1}, X_{2}) ≅ X_{1}X_{2}, let k(φ) = k(t) for an indeterminate t.

13We write φ ⊗ K′ instead of K′ ⊗ φ in order to be in harmony with the notation in §3 and in the literature. {Many authors more briefly write φ_{K′}. Starting with §16 we occasionally will do this too.}

§9 Generic Splitting of Regular Quadratic Forms 71

We want to show that k(φ) is a generic zero field of φ. This is very simple when φ is isotropic: Obviously k itself is a generic zero field of φ in this case. We thus have to show that k(φ) is specialization equivalent to k over k, which now means that there exists a place from k(φ) to k over k.

Lemma 1. If φ is an isotropic form over k, then k(φ) is a purely transcendental field extension of k.

Proof. This is by definition clear when dim φ = 2, i.e. when φ ≅ H. Now let n: = dim φ > 2. We have a decomposition φ ≅ H ⊥ ψ and may suppose without loss of generality that φ = H ⊥ ψ. So, if X_{1}, …, X_{n} are indeterminates, we have

φ(X_{1}, …, X_{n}) = X_{1}X_{2} + ψ(X_{3}, …, X_{n}).

Therefore,^{14}

k(φ) = Quot(k[X_{1}, …, X_{n}]/(X_{1}X_{2} + ψ(X_{3}, …, X_{n})) = k(x_{1}, …, x_{n}),

where x_{i} of course denotes the image of X_{i} in k(φ). The elements x_{2}, x_{3}, …, x_{n} are algebraically independent over k and x_{1} = −x_{2}^{−1}ψ(x_{3}, …, x_{n}). Therefore, k(φ) = k(x_{2}, …, x_{n}) is purely transcendental over k.

Since k(φ)/k is purely transcendental, there are many places from k(φ) to k over k, cf. [Bo_{2}, §10, Prop. 1].

We will move on to prove that also for anisotropic φ with dim φ ≥ 2, k(φ) is a generic zero field of φ. De facto we will obtain a stronger result (Theorem 4 below), and we will need its full strength later on as well. We require a lemma about the extension of places to quadratic field extensions.

Lemma 2. Let E be a field, K a quadratic extension of E and α a generator of K over E. Let p(T) = T^{2} − aT + b ∈ E[T] be the minimal polynomial of α over E. Furthermore, let ρ: E → L ∪ ∞ be a place with ρ(b) ≠ ∞ and ρ(a) ≠ ∞, ρ(a) ≠ 0. Finally, let β be an element of L such that

β^{2} − ρ(a)β + ρ(b) = 0.

Then there exists a unique place λ: K → L ∪ ∞ with λ(α) = β, which extends ρ.

Proof. Let 𝔬 denote the valuation ring of ρ, 𝔪 its maximal ideal and k the residue class field 𝔬/𝔪. Let γ: E → k ∪ ∞ be the canonical place associated to 𝔬. Then ρ = \overline{ρ} ∘ γ, where \overline{ρ}: k ↪ L is a field embedding. We may assume without loss of generality that k is a subfield of L and that \overline{ρ} is the inclusion map k ↪ L. Every place λ: K → L ∪ ∞ which extends ρ, has as image a field which is algebraic over k. Therefore we may replace L by the algebraic closure of k in L and thus assume without loss of generality that L is algebraic over k. If c is an element of 𝔬, then \overline{c} will denote the image of c in 𝔬/𝔪 = k, i.e. \overline{c} = γ(c). We have β^{2} − \overline{a}β + \overline{b} = 0.

14If A is an integral domain, QuotA denotes the quotient field of A.

By the general extension theorem for places [Bo_{2}, §2, Prop. 3], there exists a place from K to the algebraic closure k̃ of k, which extends γ. We choose such a place δ: K → k̃ ∪ ∞. Let 𝔒 be the valuation ring of K, associated to δ, 𝔐 its maximal ideal and F: = 𝔒/𝔐 its residue class field. Finally, let σ: K → F ∪ ∞ be the canonical place of 𝔒. Then σ extends the place γ: E → k∪∞. By general valuation theory [Bo_{2}, §8, Th. 1], we have [F: k] ≤ 2. We may envisage F and L as subfields of k̃, which both contain k. One of the things we will show, is that F is equal to the subfield L′: = k(β) of L.

The field extension K/L is separable since the coefficient a in the minimal polynomial p(T) is different from zero. Let j be the involution of K over E, i.e. the automorphism of K with fixed field E.

Case 1: L′ ≠ k. So β ∉ k and T^{2} − \overline{a}T + \overline{b} is the minimal polynomial of β over k. Now α^{2} − αa + b = 0 implies σ(α)^{2} − σ(α)\overline{a} + \overline{b} = 0. Thus σ(α) = β or σ(α) = \overline{a} − β. In particular, L′ = k(σ(α)) ⊂ F. Since [L′: k] = 2 and [F: k] ≤ 2, we have L′ = F and F ⊂ L.

We have j(α) = a−α, and thus σj(α) = \overline{a}−σ(α). Since \overline{a} ≠ 0 this implies σj(α) ≠ σ(α). Therefore the places σ and σ ∘ j from K to L are different. They both extend the place ρ. By general valuation theory [Bo_{2}, §8, Th. 1], there can be no other places from K to k̃ which extend ρ. Now take λ = σ in case σ(α) = β, and λ = σ ∘ j in case σ(α) = \overline{a} − β. Then λ: K → L ∪ ∞ is the only place which extends ρ and maps α to β.

Case 2: L′ = k. This time β ∈ k. Since β is a root of T^{2} − \overline{a}T + \overline{b}, we have T^{2} − \overline{a}T + \overline{b} = (T − β)(T − \overline{a} + β) and α^{2} − α\overline{a} + \overline{b} = 0 implies again that σ(α)^{2} − σ(α)\overline{a} + \overline{b} = 0. Therefore,

(σ(α) − β)(σ(α) − \overline{a} + β) = 0.

Hence σ(α) = β or σ(α) = \overline{a} − β. In particular we have σ(α) ∈ k.

Again the places σ and σ ∘ j are different. By general valuation theory, they are exactly all the places from K to k̃ which extend ρ and [F: k] = 1, i.e. F = k. As in Case 1, we take λ = σ in case σ(α) = β, and λ = σ ∘ j in case σ(α) = \overline{a} − β. Again λ is the only place from K to L (even the only place from K to k̃) which extends ρ and maps α to β.

Theorem 3. Let λ: K → L ∪ ∞ be a place and φ a form over K with GR with respect to λ and dim φ ≥ 2. Let \overline{φ}: = λ_{∗}(φ). Then λ can be extended to a place μ: K(φ) → L(\overline{φ}) ∪ ∞.

Proof. If φ is isotropic, then K(φ)/K is purely transcendental by Theorem 1. In this case λ can be extended in many ways to a place from K(φ) to L. So suppose that φ is anisotropic.

Let n: = dim φ and let 𝔬 be the valuation ring of λ. If n = 2 we assume in addition that \overline{φ} ≅ H̸, deferring the case \overline{φ} ≅ H to the end of the proof. After a linear transformation of K^{n} we may assume without loss of generality that φ is of the form (∗) as in the beginning of this section (thus with a_{i}, b_{i} ∈ 𝔬, c ∈ 𝔬^{∗} etc.). We denote by \overline{φ} the form over L, obtained from φ by replacing the coefficients a_{i}, b_{i} and – if n is odd – c by \overline{a}_{i}: = λ(a_{i}), \overline{b}_{i}: = λ(b_{i}), \overline{c}: = λ(c). Let X_{1}, …, X_{n}, resp. U_{1}, …, U_{n} be indeterminates, then we also write

φ(X_{1}, …, X_{n}) = a_{1}X_{1}^{2} + X_{1}X_{2} + b_{1}X_{2}^{2} + ψ(X_{3}, …, X_{n}),

and accordingly

\overline{φ}(U_{1}, …, U_{n}) = \overline{a}_{1}U_{1}^{2} + U_{1}U_{2} + \overline{b}_{1}U_{2}^{2} + \overline{ψ}(U_{3}, …, U_{n}).

Furthermore we write

K(φ) = Quot\frac{K[X_{1}, …, X_{n}]}{(φ(X_{1}, …, X_{n}))} = K(x_{1}, …, x_{n}),

L(\overline{φ}) = Quot\frac{L(U_{1}, …, U_{n})}{(\overline{φ}(U_{1}, …, U_{n}))} = L(u_{1}, …, u_{n}),

where x_{i} denotes of course the image of X_{i} in K(φ) and u_{i} the image of U_{i} in L(\overline{φ}). We then have the relations

\begin{matrix}a_{1}x_{1}^{2} + x_{1}x_{2} + b_{1}x_{2}^{2} + ψ(x_{3}, …, x_{n}) = 0, \\ \overline{a}_{1}u_{1}^{2} + u_{1}u_{2} + \overline{b}_{1}u_{2}^{2} + \overline{ψ}(u_{3}, …, u_{n}) = 0. \end{matrix}

{If n = 2, the last summands on the left should be read as zero.} It is easy to see that the space N: = [{a_{1} \atop 1} {1 \atop b_{1}} ] over 𝔬 is isometric to a space [{a′_{1} \atop 1} {1 \atop b′_{1}} ] with a′_{1} ∈ 𝔬^{∗}. {Lift a suitable basis of N/𝔪N to a basis of N.} Therefore we additionally assume, without loss of generality, that a_{1} ∈ 𝔬^{∗}, i.e. \overline{a}_{1} ≠ 0.

The elements x_{2}, …, x_{n} are algebraically independent over K and likewise the elements u_{2}, …, u_{n} are algebraically independent over L. Let

E: = K(x_{2}, …, x_{n}) ⊂ K(φ)

and

F: = L(u_{2}, …, u_{n}) ⊂ L(\overline{φ}).

Our place λ has exactly one extension λ̃: E → F ∪ ∞ with λ̃(x_{i}) = u_{i} (2 ≤ i ≤ n) (cf. [Bo_{2}, §10, Prop. 2]).

Let 𝔬̃ be the valuation ring of λ̃. We have K(φ) = E(x_{1}), L(\overline{φ}) = F (u_{1}) with relations

x_{1}^{2} + ax_{1} + b = 0, u_{1}^{2} + \overline{a}u_{1} + \overline{b} = 0,

where a: = a_{1}^{−1}x_{2}, b: = a_{1}^{−1}(b_{1}x_{2}^{2} + ψ(x_{3}, …, x_{n})) are in 𝔬̃ and \overline{a} = λ̃(a), \overline{b} = λ̃(b).

Surely K(φ) ≠ E, because φ ⊗ K(φ) is isotropic, but φ ⊗ E is anisotropic since the extension E/K is purely transcendental. Therefore K(φ) is a quadratic field extension of E. Furthermore \overline{a} = \overline{a}_{1}^{−1}u_{2} ≠ 0. Hence we can apply Lemma 2 to the place λ̃: E → F ∪ ∞ and the field extensions K(φ)/E, L(\overline{φ})/F. According to this lemma, there exists a (unique) place μ: K(φ) → L(\overline{φ}) which extends λ̃ and maps x_{1} to u_{1}. This proves the theorem in case \overline{φ} ≅ H̸.

Finally, let n = 2, φ anisotropic, but \overline{φ} isotropic. We can still use the description of K(φ) given above (this time with ψ = 0), i.e. φ = [{a_{1} \atop 1} {1 \atop b_{1}} ] with a_{1} ∈ 𝔬^{∗}, b_{1} ∈ 𝔬, K(φ) = K(x_{1}, x_{2}) with x_{2} transcendental over K and

(1)  a_{1}x_{1}^{2} + x_{1}x_{2} + b_{1}x_{2}^{2} = 0.

We want to extend λ to a place μ: K(φ) → L ∪ ∞. In order to do this, we choose a nontrivial zero (c_{1}, c_{2}) ∈ L^{2} of \overline{φ}. Then we choose an extension λ̃: K(x_{2}) → L ∪ ∞ of λ with λ̃(x_{2}) = c_{2}, cf. [Bo_{2}, §10, Prop. 1]. {Take the “variable” x_{2} − c_{2} there.} We have

(2)  \overline{a}_{1}c_{1}^{2} + c_{1}c_{2} + \overline{b}_{1}c_{2}^{2} = 0.

If c_{2} = 0, then c_{1} = 0 by (2). Our zero is nontrivial however, so c_{2} ≠ 0.

Equation (1) shows that x_{1}^{2}+ax_{1}+b = 0 with a: = a_{1}^{−1}x_{2}, b: = a_{1}^{−1}b_{1}x_{2}^{2}. We have λ̃(a) = \overline{a}_{1}^{−1}c_{2} ≠ 0, λ̃(b) = \overline{a}_{1}^{−1}\overline{b}_{1}c_{2}^{2}. Equation (2) shows that c_{1}^{2}+λ̃(a)c_{1}+ λ̃(b) = 0. Since φ is anisotropic, K(φ) ≠ K(x_{1}), thus [K(φ): K(x_{1})] = 2. Lemma 2 tells us that there exists a (unique) place μ: K(φ) → L ∪ ∞ which extends λ̃.

Remark. In the proof we did not need the uniqueness statement of Lemma 2. Nonetheless, it deserves some attention. For example, one can use it to deduce from our proof that, given an anisotropic \overline{φ}, there is exactly one place μ: K(φ) → L(\overline{φ}) ∪ ∞ which extends λ and maps x_{i} to u_{i} (1 ≤ i ≤ n). Now, this holds for the generators x_{1}, …, x_{n} of K(φ) and u_{1}, …, u_{n} of L(\overline{φ}), associated to the special representation (∗) of φ above (still with a_{1} ∈ 𝔬^{∗}) and can, by means of a coordinate transformation (with coefficients in 𝔬, etc.), be transferred to the case where φ = [a_{ij}] for an arbitrary symmetric matrix (a_{ij}) over 𝔬 with

det ( \begin{matrix}2a_{11} & … & a_{1n} \\ ⋮ & & ⋮ \\ a_{n1} & … & 2a_{nn} \end{matrix} ) ∈ 𝔬^{∗}.

Theorem 4. Let λ: K → L ∪ ∞ be a place. Let φ be a form over k with dim φ ≥ 2, which has GR with respect to λ. Then λ_{∗}(φ) is isotropic if and only if λ can be extended to a place μ: K(φ) → L ∪ ∞.

Proof. If λ can be extended to a place μ from K(φ) to L, then λ_{∗}(φ) has to be isotropic by the standard argument, which we already used in §4, just after Theorem 2.

Conversely, suppose that \overline{φ}: = λ_{∗}(φ) is isotropic. By Theorem 3, there exists a place

μ: K(φ) → L(\overline{φ}) ∪ ∞

which extends λ. By Lemma 1, L(\overline{φ})/L is a purely transcendental extension. Therefore there exists a place

ρ: L(\overline{φ}) → L ∪ ∞

over L. Now ρ ∘ μ: K(φ) → L ∪ ∞ is a place which extends λ.

Remark. We could have adapted the proof of Theorem 3 in such a way, that we would have obtained Theorem 4 immediately (see in particular our argument there in the case \overline{φ} ≅ H). On the other hand, we can obtain Theorem 3 from Theorem 4 by applying the latter to the place

j ∘ λ: K(φ) → L(\overline{φ}) ∪ ∞,

where j is the inclusion L ↪ L(\overline{φ}). For our further investigations (§10, §12, §13) it is better however, to isolate Theorem 3 and its proof as a stopover to Theorem 4.

Corollary. Let φ be a form over a field k with dim φ ≥ 2. Then k(φ) is a generic zero field of φ.

Proof. Clearly φ ⊗ k(φ) is isotropic. Now let L ⊃ k be a field extension with φ ⊗ L isotropic. Applying Theorem 4 to the trivial place k ↪ L, we see that there exists a place μ: k(φ) → L ∪ ∞ over k.

Since Theorems 1 and 2 of §4 are subcases of our current Theorem 4, they are now proved. The definitions and theorems following these two theorems remain valid. Later on they will be used without further comments in renewed generality, namely for arbitrary characteristic instead of characteristic ≠ 2, regular quadratic forms instead of the nondegenerate symmetric bilinear forms which occur there. In particular, for every regular quadratic form φ over k, we have a generic splitting tower (K_{i} | 0 ≤ i ≤ h) with higher indices i_{r} and higher kernel forms φ_{r}.

Comments. It should be noted that I. Kersten and U. Rehmann have used a different route in [Ke R, §6] to construct, for every form φ over k and every r ≤ [\frac{dim φ}{2}], a field extension of k which is generic for the splitting off of r hyperbolic planes. In particular one can already find a generic splitting tower of φ in their work. In [KR] the generic splitting of regular quadratic forms in arbitrary characteristic is based on these foundations.

The results about forms of height 1 and 2, cited at the end of our §4, have so far only been established for characteristic ≠ 2 in the literature.

76 Chapter II. Generic Splitting Theory

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§10 Separable Splitting

All fields occurring in this section are supposed to have characteristic 2. If φ is a nondegenerate quadratic form over such a field k, then its quasilinear part (cf. §6, Definition 1), which we denote by QL(φ), is anisotropic. Now, if K ⊃ k is a field extension, then it can happen that QL(φ ⊗ K) = QL(φ) ⊗ K is isotropic, and thus that φ ⊗ K is degenerate. This possibility prompted us, when dealing with the theory of generic splitting in the previous sections, to only allow regular forms, i.e. quadratic forms φ with dim QL(φ) ≤ 1.

We will now make a course change and allow arbitrary nondegenerate quadratic forms. We will however only tolerate a restricted class of field extensions, the so-called separable field extensions. We will see that a reasonably satisfying theory of generic splitting is still possible in this case.

A field extension K ⊃ k is called separable if every finitely generated subextension E ⊃ k has a separating transcendence basis, i.e. a transcendence basis t_{1}, …, t_{n}, such that E is separably algebraic over k(t_{1}, …, t_{n}), cf. [Bo_{3}], [Lg_{1}, X, §6], [J, IV, §5]. If K is already finitely generated over k, then it suffices to check if K itself contains a separating transcendence basis over k (loc. cit.).

In what follows, a “form” will always be understood to be a quadratic form. The foundation for the rest of this section is

Theorem 1. Let φ be a nondegenerate form over k, and let K ⊃ k be a separable field extension. Then φ ⊗ K is also nondegenerate.

Proof. We work in the algebraic closure K̃ of K. We have QL(φ) = [a_{1}, …, a_{r}] with a_{i} ∈ K. This form is anisotropic. We have to show that QL(φ) ⊗ K is also anisotropic. Each a_{i} has exactly one square root \sqrt{a_{i}} ∈ K̃, which is already in the radical closure k^{1/2^{∞}} ⊂ K^{1/2^{∞}} ⊂ K̃ of k. Now, since QL(φ) is anisotropic, the elements a_{1}, …, a_{n} are linearly independent over the subfield k^{2} = \{x^{2} | x ∈ k\} of k, or equivalently, the elements \sqrt{a_{1}}, …, \sqrt{a_{n}} are linearly independent over k. By an important theorem about separable field extensions (“MacLane’s Criterion”, loc. cit.), the fields K and k^{1/2^{∞}} are linearly disjoint over k. Therefore, the elements \sqrt{a_{1}}, …, \sqrt{a_{r}} are linearly independent over K. This shows that QL(φ) ⊗ K is anisotropic.

If 𝔬 is a valuation ring with maximal ideal 𝔪, we denote the residue class field 𝔬/𝔪 from now on by κ(𝔬).

Theorem 2. Let λ: K → L ∪ ∞ be a place and φ a form which has GR with respect to λ.^{15} Suppose that the form λ_{∗}(φ) is nondegenerate. Suppose further that K′ ⊃ K is a field extension and that μ: K′ → L ∪ ∞ is an extension of the place λ. Then the form φ ⊗ K′ has GR with respect to μ and μ_{∗}(φ ⊗ K′) ≅ λ_{∗}(φ).

Proof. Let 𝔬: = 𝔬_{λ}, 𝔬′: = 𝔬_{μ}. The field extension \overline{λ}: κ(𝔬) ↪ L is a combination of the extensions κ(𝔬) ↪ κ(𝔬′) and \overline{μ}: κ(𝔬′) ↪ L, where the first extension is induced by the inclusion 𝔬 ↪ 𝔬′.

Let E be a quadratic space for φ and M a nondegenerate quadratic 𝔬module with E ≅ K ⊗_{𝔬} M. Then K′ ⊗ E = K′ ⊗_{𝔬′} M′ with M′: = 𝔬′ ⊗_{𝔬} M. The quasilinear quadratic κ(𝔬)-module G: = κ(𝔬) ⊗_{𝔬} QL(M) is anisotropic. By assumption, L ⊗\overline{λ} G = QL(L ⊗_{λ} M) is also anisotropic. Therefore

κ(𝔬′) ⊗_{κ(𝔬)} G = κ(𝔬′) ⊗_{𝔬′} QL(M′)

is anisotropic. This proves that M′ is a nondegenerate quadratic 𝔬′-module. Hence φ ⊗ K′ is nondegenerate and has GR with respect to μ. Furthermore, μ_{∗}(φ ⊗ K′) corresponds to the quadratic space

L ⊗_{μ} M′ = L ⊗_{μ} (𝔬′ ⊗_{𝔬} M) = L ⊗_{λ} M.

Hence μ_{∗}(φ ⊗ K′) ≅ λ_{∗}(φ).

Now we can faithfully repeat the observation, made at start of our treatment of the theory of generic splitting in §4.

So, let φ be a nondegenerate form over a field k (of characteristic 2), and let K ⊃ k and L ⊃ k be field extensions of k with L ⊃ k separable. Let λ: K → L ∪ ∞ be a place over k. On the basis of Theorem 1, we can apply Theorem 2 to the trivial place k ↪ L and its extension λ. We see that φ ⊗ K is nondegenerate and has GR with respect to λ, and that λ_{∗}(φ⊗K) ≅ φ⊗L.

Let φ ⊗ K ≅ φ_{1} ⊥ r_{1} × H be the Witt decomposition of φ. By an established argument (§8, Theorem 2(b)), φ_{1} has GR with respect to λ, and φ ⊗ L ≅ λ_{∗}(φ_{1}) ⊥ r_{1} × H. Hence, ind(φ ⊗ L) ≥ ind(φ ⊗ K). If K is also separable over k and if K and L are specialization equivalent over k, it follows again that ind(φ ⊗ L) = ind(φ ⊗ K) and ker(φ ⊗ L) ≅ λ_{∗}(φ_{1}).

Definition. A generic separable zero field of φ is a separable field extension K ⊃ k which has the following properties:

(a) φ ⊗ K is isotropic.

(b) If L ⊃ k is a separable field extension with φ ⊗ L isotropic, then there exists a place λ: K → L ∪ ∞ over k.

According to Theorem 1, φ can become isotropic over a separable field extension L of k only if φ ≠ QL(φ), i.e. if dim φ − dim QL(φ) ≥ 2. If φ

15This assumption presupposes that φ is nondegenerate (cf. §7, Definition 1).

§10 Separable Splitting 79

is such a form with dim φ ≠ 2, then the polynomial φ(X_{1}, …, X_{n}) is irreducible over the algebraic closure k̃ of k. This can easily be seen by writing φ(X_{1}, …, X_{n}) = X_{1}X_{2} + ψ(X_{3}, …, X_{n}) over k̃, after a coordinate transformation. Here ψ is a quadratic polynomial which is not the zero polynomial. Therefore, we can again construct the field

k(φ) = Quot\frac{k[X_{1}, …, X_{n}]}{(φ(X_{1}, …, X_{n}))}.

This extends our definition of k(φ) in §9 from regular forms φ to nondegenerate forms φ. {Obviously k(φ) will have its original meaning when dim φ = 2, QL(φ) = 0.}

It is now easy to see that k(φ) is separable over k: Let x_{1}, …, x_{n} be the images of X_{1}, …, X_{n} in k(φ), i.e. k(φ) = k(x_{1}, …, x_{n}). After a coordinate transformation we may suppose, without loss of generality, that

φ(X_{1}, …, X_{n}) = a_{1}X_{1}^{2} + X_{1}X_{2} + b_{1}X_{2}^{2} + ψ(X_{3}, …, X_{n}).

The elements x_{2}, …, x_{n} form a transcendence basis of k(φ) over k. If x_{1} ∉ k(x_{2}, …, x_{n}), then k(φ) is a separable quadratic extension of k(x_{2}, …, x_{n}). If the form φ is isotropic, we can make it so that a_{1} = 0. Then k(φ) = k(x_{2}, …, x_{n}) is purely transcendental over k.

Theorem 3. Let φ be a nondegenerate form over a field K with φ ≠ QL(φ) and let λ: K → L ∪ ∞ be a place such that φ has GR with respect to λ. Suppose that also the form λ_{∗}(φ) is nondegenerate. Then λ_{∗}(φ) is isotropic if and only if λ can be extended to a place μ: K(φ) → L ∪ ∞.

Proof. If there exists such a place μ, then λ_{∗}(φ) is isotropic by an established argument, using Theorem 2 above.

Suppose now that \overline{φ}: = λ_{∗}(φ) is isotropic. By the theory in §9, we may suppose that dim QL(φ) ≥ 2, thus dim φ ≥ 4. Just as in the proof of Theorem 3 in §9, we see that λ can be extended to a place μ: K(φ) → L(\overline{φ}) ∪ ∞. \{Note that this is also true in case \overline{φ} is anisotropic.\} Since \overline{φ} is isotropic, L(\overline{φ}) is a purely transcendental extension of L, as established above. Hence there exists a place ρ from L(\overline{φ}) to L over L. The place ρ ∘ μ: K(φ) → L ∪ ∞ extends λ.

Corollary. Let φ be a nondegenerate form over a field k with φ ≠ QL(φ). Then k(φ) is a generic separable zero field of φ.

Since we have secured the existence of a generic separable zero field, we obtain for every form φ over k a generic separable splitting tower

(K_{r} | 0 ≤ r ≤ h)

with higher indices i_{r} and higher kernel forms φ_{r} (0 ≤ r ≤ h), in complete analogy with the construction of generic splitting towers in §4 (just before §4, Definition 5). Thus K_{0}/k is a separable inessential field extension, φ_{r} = ker(φ ⊗ K_{r}), and K_{r+1} is a generic separable zero field of φ_{r} for r < h. The height h satisfies h ≤ \frac{1}{2}(dim φ − dim QL(φ)).

In analogy with §4, Theorem 3, we have the following theorem, with mutatis mutandis the same proof.

Theorem 4. Let φ be a nondegenerate form over k. Let (K_{r} | 0 ≤ r ≤ h) be a generic separable splitting tower of φ with associated higher kernel forms φ_{r} and indices i_{r}. Let γ: k → L ∪ ∞ be a place such that φ has GR with respect to γ. Suppose that the form γ_{∗}(φ) is nondegenerate. Finally, for an m with 0 ≤ m ≤ h, let a place λ: K_{m} → L ∪ ∞ be given, which extends γ and which cannot be extended to K_{m+1} in case m < h. Then φ_{m} has GR with respect to λ. The form γ_{∗}(φ) has kernel form λ_{∗}(φ_{m}) and Witt index i_{0} + … + i_{m}.

Now we can faithfully repeat Scholium 1 to 4 from §4, but this time with generic separable splitting towers and separable field extensions. We leave this to the reader. {In Scholium 3, on should of course assume – as in Theorem 4 above – that γ_{∗}(φ) is nondegenerate.}

In particular, the generic separable splitting tower (K_{r} | 0 ≤ r ≤ h) regulates the splitting behaviour of φ with respect to separable field extensions L ⊃ k, as described in §4, Scholium 1.

§11 Fair Reduction and Weak Obedience

Our specialization theory of quadratic forms, developed in §6 – §8, gave a satisfying basis for understanding the splitting behaviour of quadratic forms under field extensions (if the forms were not regular, we had to limit ourselves to separable field extensions). There is one important point however, where the specialization theory is disappointing.

For instance, let λ: K → L∪∞ be a place from a field K of characteristic 0 to a field L of characteristic 2. If φ is a quadratic form over K which has good reduction with respect to λ, then λ_{∗}(φ) is automatically a strictly regular form. Conversely, given a strictly regular form ψ over L, one can easily find a strictly regular (= nondegenerate) form φ over K which has good reduction with respect to λ and such that λ_{∗}(φ) ≅ ψ. One could then try to deduce properties of ψ from properties of the “lifting” φ of ψ, in the hope that φ is easier to deal with than ψ because char K = 0. Good examples can be obtained from the generalization of §4, Scholium 3, at the end of §9, but we will not carry this out.

It is furthermore desirable to lift a nondegenerate form ψ over L, with quasilinear part QL(ψ) ≠ 0, to a form φ over K. This is not possible with our specialization theory as it stands.

Let us review the foundations of the current theory! We return to the use of geometric language. Let λ: K → L∪∞ be a place with associated valuation ring 𝔬. Let 𝔪 be the maximal ideal of 𝔬 and k = 𝔬/𝔪 its residue class field. Finally, let \overline{λ}: k ↪ L be the field extension determined by λ.

Given a suitable quadratic space E over K, the idea was to attach a space F over k to E and then to define the specialization λ_{∗}(E) of E with respect to λ as the space L ⊗\overline{λ} F. We required E to have good reduction, i.e. E ≅ K ⊗_{𝔬} M for a nondegenerate quadratic module M over 𝔬. Then we could choose our space F over k to be F = M/𝔪M.

The preceding is analogous to the specialization theory for varieties in algebraic and arithmetic geometry (and related areas, such as rigid analysis): One defines a class of “nondegenerate” objects X over the valuation ring 𝔬 and associates to it, by means of a base extension, the objects K ⊗_{𝔬} X and k ⊗_{𝔬} X over K and k. Finally one decrees that L ⊗\overline{λ} (k ⊗_{𝔬} X) = L ⊗_{λ} X is the specialization of K ⊗_{𝔬} X with respect to λ.

Without any doubt, we have found in the nondegenerate quadratic spaces, as defined in §6, Definition 3, a respectable class X of such objects which easily comes to mind. Nevertheless we are a bit unfair to all the quadratic spaces over K, waiting to be specialized. It would be fairer to only require that E ≅ K ⊗_{𝔬} M with M a free quadratic 𝔬-module such that M/𝔪M is nondegenerate, and then to define λ_{∗}(E): = L ⊗_{λ} M = L ⊗\overline{λ} (M/𝔪M). But then one should show, that up to isometry, λ_{∗}(E) only depends on λ and E and not on the choice of M.

This is possible, as we shall show now. In the following a quadratic module over a valuation ring (in particular a field) will always be understood to be a free quadratic module of finite rank.

Let 𝔬 be a valuation ring with maximal ideal 𝔪, quotient field K and residue class field k = 𝔬/𝔪. If 2 ∉ 𝔪, then we already know all what follows. On formal grounds, we allow nevertheless the uninteresting case that char k ≠ 2.

Definition 1.

(a) A quadratic module M over 𝔬 is called reduced nondegenerate if the quadratic module M/𝔪M over k is nondegenerate. (Note: If char k ≠ 2, this implies that M itself is nondegenerate.)

(b) A quadratic module E over K has fair reduction (or: FR for short) with respect to λ (or: with respect to 𝔬), if there exists a reduced nondegenerate quadratic module M over 𝔬 such that E ≅ K ⊗_{𝔬} M.^{16}

It is our task to prove that in the situation of Definition 1(b), the space M/𝔪M is independent of the choice of reduced nondegenerate module M up to isometry. We will however proceed in a more general direction than necessary, in order to develop at the same time the equipment necessary to obtain a generalization of the important Theorem 2(b) (§8) for fair reduction instead of good reduction. First a very general definition.

Definition 2. Let M = (M, q) and M′ = (M′, q′) be quadratic modules over a ring A. We say that M represents the quadratic module M′, and write M′ < M, if the A-module M has a direct sum decomposition M = M_{1} ⊕ M_{2} with (M_{1}, q|M_{1}) ≅ (M′, q′).

Now, if M is a quadratic 𝔬-module, we always regard M as an 𝔬-submodule of the K-vector space K ⊗_{𝔬} M. So we have K ⊗_{𝔬} M = KM. In the following, we will almost always denote the quadratic form on M by q, and its associated bilinear form B_{q} by B. We will also use q to denote the quadratic form on K ⊗_{𝔬} M with values in K, obtained from q. We will often denote the module M/𝔪M over k by \overline{M}, and the image of a vector x ∈ M in \overline{M} with \overline{x}. Likewise we denote the image of a scalar a ∈ 𝔬 in k by \overline{a}. Finally, we denote by \overline{q} the quadratic form on \overline{M} with values in k, induced by q, i.e. \overline{q}(\overline{x}) = \overline{q(x)} for x ∈ M. Furthermore, \overline{B} will stand for the associated bilinear form B\overline{q}, i.e. \overline{B}(\overline{x}, \overline{y}) = \overline{B(x, y)} for x, y ∈ M.

16In [K_{4}] the words “nearly good reduction” are used instead of “fair reduction”. We avoid this terminology here, because we will talk about a different concept, “almost good reduction” later (Chapter IV).

As before, we call a nondegenerate quadratic module over a field or valuation ring a quadratic space, or just a space. If M is a reduced nondegenerate quadratic 𝔬-module, then \overline{M} = M/𝔪M is a space over k. It is easy to show that K ⊗M is a space over K too (see the Remark below, after Definition 3), but that is not so important at the moment.

Lemma 1. Let M be a reduced nondegenerate quadratic module over 𝔬. There exists a decomposition M = M_{1} ⊥ M_{2} with M_{1} strictly regular and B(M_{2}×M_{2}) ⊂ 𝔪. If such a decomposition is given and x is a primitive vector of M_{2}, then q(x) ∈ 𝔬^{∗}.

Proof. We have a decomposition \overline{M} = U ⊥ QL(\overline{M}) with U strictly regular.^{17} Let x_{1}, …, x_{r} be vectors of M, such that the images \overline{x}_{1}, …, \overline{x}_{1} in \overline{M} form a basis of the k-vector space U. Then the determinant of the value matrix

(B(x_{i}, x_{j})) with 1 ≤ i, j ≤ r is a unit of 𝔬. Therefore, M_{1}: = ∑_{i=1}^{r} 𝔬x_{i} is a

strictly regular space. Let M_{2} be the orthogonal complement M_{1}^{⊥} of M_{1} in M. Then M = M_{1} ⊥ M_{2} and M_{2}/𝔪M_{2} = QL(\overline{M}). Since QL(\overline{M}) is quasilinear, we have B(M_{2}×M_{2}) ⊂ 𝔪. Conversely, if such a decomposition M = M_{1} ⊥ M_{2} is given, with M_{1} strictly regular and B(M_{2} × M_{2}) ⊂ 𝔪, then \overline{M}_{2} = QL(\overline{M}). Since QL(\overline{M}) is anisotropic, every primitive vector x in M_{2} has a value q(x) ∈ 𝔬^{∗}.

Lemma 2. Suppose again that M is a reduced nondegenerate quadratic 𝔬module. Then M is maximal among all finitely generated (thus free) 𝔬-modules N ⊂ K ⊗_{𝔬} M with q(N) ⊂ 𝔬.

Proof. We work in the quadratic K-module E: = K ⊗_{𝔬} M. According to Lemma 1, there is a decomposition M = M_{1} ⊥ M_{2} with M_{1} strictly regular and B(M_{2} × M_{2}) ⊂ 𝔪. Let x ∈ E and q(M + 𝔬x) ⊂ 𝔬. We have to show that x ∈ M.

We write x = x_{1} +x_{2} with x_{1} ∈ K ⊗M_{1}, x_{2} ∈ K ⊗M_{2}. For every y ∈ M_{1}, we have B(x_{1}, y) = B(x, y) = q(x + y) − q(x) − q(y) ∈ 𝔬. Since the bilinear form B = B_{q} is nondegenerate on M_{1}, it follows that x_{1} ∈ M_{1}. Therefore, M + 𝔬x = M + 𝔬x_{2} and so q(x_{2}) ∈ 𝔬. Now, x_{2} = az for a primitive vector z ∈ M_{2} and a ∈ K. Since q(z) ∈ 𝔬^{∗}, we get a ∈ 𝔬, and so x_{2} ∈ M. We conclude that x ∈ M.

Lemma 3. (Extension of §6, Lemma 4.) Let M be a reduced nondegenerate quadratic 𝔬-module and let e be a primitive isotropic vector in M. Then e can be completed to a hyperbolic vector pair e, f in M.

Proof. We work again in E = K ⊗ M. The ideal B(e, M) of 𝔬 is finitely generated. Therefore we have B(e, M) = a𝔬 for an element a of 𝔬. If a were equal to zero, then the vector \overline{e} ∈ \overline{M} would lie in the quasilinear part QL(\overline{M}) of \overline{M}. However, \overline{e} ≠ 0 and \overline{q}(\overline{e}) = 0, so that \overline{e} ∉ QL(\overline{M}) due to the anisotropy

17Recall that QL(\overline{M}) denotes the quasilinear part of \overline{M}, see the beginning of §6.

of QL(\overline{M}). Therefore, a ≠ 0. Since B(e, M) = a𝔬 and q(M) ⊂ 𝔬, we have q(M + 𝔬(a^{−1}e)) ⊂ 𝔬. Thus, by Lemma 2, a^{−1}e ∈ M. Since e is primitive, we conclude that a^{−1} ∈ 𝔬, i.e. a ∈ 𝔬^{∗}. Hence, B(e, M) = 𝔬. As in the proof of §6, Lemma 4, we choose z ∈ M with B(e, z) = 1 and complete e to a hyperbolic pair with the vector f: = z − q(z)e.

Theorem 1. Let M be strictly regular, N a reduced nondegenerate quadratic 𝔬-module and K ⊗ M < K ⊗ N. Then M < N. Furthermore, N ≅ M ⊥ P where P is a reduced nondegenerate quadratic 𝔬-module.

Proof. If Q is a strictly regular quadratic submodule of N, then N = Q ⊥ P with P: = Q^{⊥} = {x ∈ N | B(x, Q) = 0}, since Q is nondegenerate with respect to the bilinear form B = B_{q} (§5, Lemma 1). We have \overline{N} = \overline{Q} ⊥ \overline{P} and conclude that \overline{P} is nondegenerate, i.e. P is reduced nondegenerate.

Because of this preliminary remark, it suffices to show that M < N. Suppose first of all that M is hyperbolic, i.e. M ≅ r × H for some r > 0, where H denotes the quadratic module [{0 \atop 1} {1 \atop 0}] over 𝔬. We proceed by induction on r.

Since K ⊗ M < K ⊗ N, K ⊗ N contains isotropic vectors. We can then choose a primitive isotropic vector e in N. This vector can be completed to a hyperbolic vector pair in N, by Lemma 3. Hence, H < N and so N ≅ H ⊥ N′ where N′ is a reduced nondegenerate quadratic 𝔬-module. From r × (K ⊗ H) < K ⊗ N, we get K ⊗ N ≅ r × (K ⊗ H) ⊥ G for some space G over K. On the other hand, K⊗N ≅ (K⊗H) ⊥ (K⊗N′). By the Cancellation Theorem over K (§6, Theorem 2), we get (r − 1) × (K ⊗ H) ⊥ G ≅ K ⊗ N′. Therefore, (r − 1) × (K ⊗ H) < K ⊗ N′. The induction hypothesis gives us (r − 1) × H < N′ and so r × H < N.

In the general case, K ⊗ M < K ⊗ N implies that K ⊗ (M ⊥ (−M)) < K ⊗ (N ⊥ (−M)). Since M ⊥ (−M) is hyperbolic, it follows from above that M ⊥ (−M) < N ⊥ (−M), and so, N ⊥ (−M) ≅ M ⊥ (−M) ⊥ P by the remark at the beginning of the proof, where P is another reduced nondegenerate quadratic 𝔬-module. By the Cancellation Theorem over 𝔬 (§6, Theorem 2), we may conclude that N ≅ M ⊥ P, and so M < N.

Theorem 2 [K_{4}, Lemma 2.8]. Suppose that M and N are reduced nondegenerate quadratic 𝔬-modules with K ⊗ M < K ⊗ N. The spaces \overline{M} = M/𝔪M and \overline{N} = N/𝔪N over k have the following properties:

(a) \overline{M} < \overline{N}.

(b) More precisely: there exist quadratic subspaces S and T of \overline{N} with \overline{M} < S, \overline{N} = S ⊥ T, T strictly regular, dim S ≤ dim QL(\overline{M}) + dim QL(\overline{N}) + dim M.

(c) If \overline{N} is anisotropic, then M < N.

Remark. We will not need properties (b) and (c) later on. They can however be obtained in the following proof at no extra cost.

Proof of Theorem 2. (i) By Lemma 1, we can choose an orthogonal decomposition M = M_{1} ⊥ M_{2} with M_{1} strictly regular and B(M_{2} × M_{2}) ⊂ 𝔪. We have K ⊗ M_{1} < K ⊗ N. By Theorem 1, we have N ≅ M_{1} ⊥ N_{2}, where N_{2} is a reduced nondegenerate quadratic 𝔬-module. Then K ⊗ M_{1} ⊥ K ⊗ M_{2} < K ⊗ M_{1} ⊥ K ⊗ N_{2} implies K ⊗ M_{2} < K ⊗ N_{2} by the Cancellation Theorem over K (§6, Theorem 2) as usual. If we could prove the claims of the theorem for M_{2} and N_{2} instead of M and N, then they would follow immediately for M and N as well. Hence we assume now, without loss of generality, that B(M × M) ⊂ 𝔪.

(ii) If M = {0}, nothing has to be done. So suppose that M ≠ {0}. Surely k has characteristic 2. We suppose, without loss of generality that F: = K ⊗ M is a subspace of E: = K ⊗ N (instead of only: F is isomorphic to a subspace of E). Since \overline{M} is anisotropic, we have M = \{x ∈ F | q(x) ∈ 𝔬\} (as already ascertained before). Therefore, N_{1}: = N ∩ F ⊂ M. Moreover, N_{1} is a direct summand of the 𝔬-module N, since N/N_{1} is torsion free and finitely generated, and thus free. Hence N_{1} itself is also free.

Let m: = dim M. By the Elementary Divisor Theorem for valuation rings, there exist bases x_{1}, …, x_{m} and y_{1}, …, y_{m} of the free 𝔬-modules M and N_{1} with y_{i} = c_{i}x_{i}, c_{i} ∈ 𝔬 (1 ≤ i ≤ m). We suppose without loss of generality that c_{i} = 1 for 1 ≤ i ≤ s and c_{i} ∈ 𝔪 for s < i ≤ m. {It is allowed that s = 0 or s = m.}

If \overline{N} is anisotropic, then q(x) ∈ 𝔬^{∗} for every primitive vector x of N, and thus for every vector in N_{1} which is primitive in N_{1}. Hence s = m, i.e. N_{1} = M and M < N. This establishes property (c).

(iii) Let V be the image of N_{1} in \overline{N} = N/𝔪N. Since N_{1} is a direct summand of N, we may make the identification V = \overline{N_{1}}. Since B(M × M) ⊂ 𝔪, we have B(N_{1} × N_{1}) ⊂ 𝔪, thus \overline{B}(V × V) = 0. Hence V is quasilinear. We have

(1)  \overline{M} = k\overline{x}_{1} ⊕ ⋯ ⊕ k\overline{x}_{m} ≅ [a_{1}, …, a_{m}], with a_{i}: = \overline{q}(\overline{x}_{i}) ∈ k^{∗}.
(2)  V = k\overline{y}_{1} ⊕ ⋯ ⊕ k\overline{y}_{m} ≅ [a_{1}, …, a_{s}] ⊥ (m − s) × [0].

Let R: = QL(\overline{N}) be the quasilinear part of \overline{N} and V_{0}: = V ∩ R. Since R is anisotropic, V_{0} is also anisotropic. From (2) it follows that V_{0} < [a_{1}, …, a_{s}], and so

(3)  [a_{1}, …, a_{s}] ≅ V_{0} ⊥ [b_{1}, …, b_{s−t}]

for elements b_{i} ∈ k^{∗} and t: = dim V_{0}. {If t = s, the righthand side of (3) should be read as V_{0}.} From (2) and (3) we get a decomposition V = V_{0} ⊥ U, with

(4)  U ≅ [b_{1}, …, b_{s−t}] ⊥ (m − s) × [0].

Since V ∩ R = V_{0}, we have U ∩ R = \{0\}. We choose a submodule W of \overline{N} such that \overline{N} = (R + U) ⊕ W = R ⊕ U ⊕ W. Since R is the quasilinear part of \overline{N}, P: = U ⊕ W is strictly regular and \overline{N} = R ⊥ P. Let u_{1}, …, u_{m−t} be a basis of U, associated with the representation (4). P is a symplectic vector space with respect to the bilinear form \overline{B} = B\overline{q}. Hence we can complete u_{1}, …, u_{m−t} to a basis u_{1}, …, u_{m−t}, z_{1}, …, z_{m−t} of a subspace P_{1} of P with \overline{B}(u_{i}, z_{j}) = δ_{ij} (1 ≤ i, j ≤ m − t). As a quadratic space, P_{1} is of the form

P_{1} = (ku_{1} + kz_{1}) ⊥ … ⊥ (ku_{m−t} + kz_{m−t})

and P = P_{1} ⊥ T for a strictly regular space T.

For s − t < i ≤ m − t, we have ku_{i} + kz_{i} ≅ [{0 \atop 1} {1 \atop 0}]. Therefore, [a_{t+i}] < ku_{i} + kz_{i} for this i. Furthermore we have V_{0} < R and [b_{i}] < ku_{i} + kz_{i} for 1 ≤ i ≤ s − t, since b_{i} = \overline{q}(u_{i}). Putting everything together, (1) and (3) yield

\overline{M} ≅ V_{0} ⊥ [b_{1}, …, b_{s−t}] ⊥ [a_{s+1}, …, a_{m}] < R ⊥ P_{1},

and R ⊥ P = R ⊥ P_{1} ⊥ T = \overline{N}. Let S: = R ⊥ P_{1}, then \overline{M} < S and \overline{N} = S ⊥ T. Since \overline{M} = QL(\overline{M}) and R = QL(\overline{N}), we have furthermore

dim S = dim QL(\overline{N}) + 2(m − t)
= dim QL(\overline{N}) + dim QL(\overline{M}) + dim \overline{M} − 2t
≤ dim QL(\overline{N}) + dim QL(\overline{M}) + dim M.

As a special case of Theorem 2, we obtain

Corollary 1. Let M and M′ be two reduced nondegenerate quadratic 𝔬modules with K ⊗ M ≅ K ⊗ M′. Then M/𝔪M ≅ M′/𝔪M′.

Now let λ: K → L ∪ ∞ again be a place and 𝔬 its associated valuation ring. As before, we denote by \overline{λ} the field embedding k ↪ L determined by λ. By Corollary 1 it makes sense to make the following definition:

Definition 3. Let E be a space over K which has FR with respect to λ. Let M be a reduced nondegenerate quadratic 𝔬-module with E ≅ K ⊗ M. We call the quadratic L-module L ⊗_{λ} M = L ⊗\overline{λ} (M/𝔪M) (which is uniquely determined by E up to isometry) the specialization of E with respect to λ, and denote it by λ_{∗}(E).

Note. If E has good reduction with respect to λ, then λ_{∗}(E) has the old meaning.

Caution! If char L = 2, then λ_{∗}(E) can be a degenerate quadratic L-module, even if E is strictly regular. However, this does not happen when the field embedding \overline{λ} is separable, see §10.

Remark. If M is a reduced nondegenerate quadratic 𝔬-module, then the quadratic K-module K ⊗ M is definitely nondegenerate, for we know that QL(K ⊗ M) = K ⊗ QL(M), and if QL(K ⊗ M) were isotropic, then QL(M), and thus QL(M/𝔪M), would also be isotropic. Therefore our assumption in Definition 3, that the quadratic K-module E is nondegenerate, is a natural one and does not cause a loss of generality.

We illustrate the concept of fair reduction with a few examples.

Example 1. Let 𝔬 be a valuation ring with char K = 0, char k = 2, and let λ: K → k ∪ ∞ be the canonical place of 𝔬. We want to lift an arbitrary space S over k to a space over K by means of λ. We choose a decomposition

S ≅ [{α_{1} \atop 1} {1 \atop β_{1}} ] ⊥ … ⊥ [{α_{m} \atop 1} {1 \atop β_{m}} ] ⊥ [γ_{1}, …, γ_{r}]

with α_{i}, β_{i}, γ_{j} ∈ k. {Since the quasilinear part of S is anisotropic, the elements γ_{1}, …, γ_{r} are linearly independent over k^{2}.} Next we choose pre-images a_{i}, b_{i}, c_{j} of the elements α_{i}, β_{i}, γ_{j} in 𝔬. Suppose that r = 2s is even. We choose elements t_{1}, …, t_{s} ∈ 𝔪. Then the K-space

E: = [{a_{1} \atop 1} {1 \atop b_{1}} ] ⊥ … ⊥ [{a_{m} \atop 1} {1 \atop b_{m}} ] ⊥ [{c_{1} \atop t_{1}} {t_{1} \atop c_{2}} ] ⊥ … ⊥ [{c_{2s−1} \atop t_{s}} {t_{s} \atop c_{2s}} ]

clearly has FR with respect to λ and λ_{∗}(E) ≅ S. If r = 2s + 1 is odd, we again choose elements t_{1}, …, t_{s} ∈ 𝔪. This time the K-space

E: = [{a_{1} \atop 1} {1 \atop b_{1}} ] ⊥ … ⊥ [{a_{m} \atop 1} {1 \atop b_{m}} ] ⊥ [{c_{1} \atop t_{1}} {t_{1} \atop c_{2}} ] ⊥ … ⊥ [{c_{2s−1} \atop t_{s}} {t_{s} \atop c_{2s}} ] ⊥ [c_{2s+1}]

has FR with respect to λ and λ_{∗}(E) ≅ S. In the special case that all t_{i} = 0, we obtain a K-space

F: = [{a_{1} \atop 1} {1 \atop b_{1}} ] ⊥ … ⊥ [{a_{m} \atop 1} {1 \atop b_{m}} ] ⊥ [c_{1}, …, c_{r}]

for every r. Since char K = 0, we can interpret F (and more generally E) as a bilinear space,

F ≅ ({2a_{1} \atop 1} {1 \atop 2b_{1}} ) ⊥ … ⊥ ({2a_{m} \atop 1} {1 \atop 2b_{m}} ) ⊥ ⟨2c_{1}, …, 2c_{r}⟩.

The elements a_{1}, …, a_{m} can always be chosen to be ≠ 0 and we obtain the diagonalization

F ≅ ⟨2a_{1}, d_{1}, …, 2a_{m}, d_{m}, 2c_{1}, …, 2c_{r}⟩

with

d_{i}: = \frac{4a_{i}b_{i} − 1}{2a_{i}} (1 ≤ i ≤ r).

Example 2. Let k be an imperfect field of characteristic 2, 𝔬 the power series ring k[[t]] in one variable t, and so K = k((t)). Choose c ∈ k \ k^{2} and let E be the space [{1 \atop 1} {1 \atop ct^{−2}} ] over K.

Claim. E has FR, but does not have GR with respect to 𝔬.

Proof. We have E ≅ K ⊗ M, where M is the quadratic 𝔬-module [{1 \atop t} {t \atop c}]. Its reduction \overline{M} = M/𝔪M is the space [{1 \atop 0} {0 \atop c}] = [1, c] over k, which is anisotropic. Therefore M is reduced nondegenerate and E has FR.

Suppose for the sake of contradiction that E ≅ K ⊗ N where N is a quadratic space over 𝔬. Since E does not have a quasilinear part, the same is true for N. Hence, N ≅ [{α \atop 1} {1 \atop β} ] with α, β ∈ 𝔬. We then have [{1 \atop 1} {1 \atop ct^{−2}} ] ≅ [{α \atop 1} {1 \atop β} ]

over K. An inspection of the Arf-invariants shows that ct^{−2} = αβ + x^{2} + x for some x ∈ k((t)). We must have x ≠ 0. We write x = ∑_{i≥d} a_{i}t^{i} for some

d ∈ ℤ and a_{d} ≠ 0. Since αβ ∈ 𝔬, we have d = −1 and c = a_{−1}^{2}. This is a contradiction since c is not a square in k. We conclude that E does not have GR.

Example 3. Again let 𝔬 = k[[t]] and k a field of characteristic 2. In §7 we determined that the space E: = [{1 \atop 1} {1 \atop t^{−1}} ] does not have GR with respect to 𝔬. Could it be that E has at least FR with respect to 𝔬?

Let us assume that this is so. Then there exists a reduced nondegenerate quadratic 𝔬-module M with E ≅ K ⊗ M. We choose a decomposition M = M_{1} ⊥ M_{2} with M_{1} strictly regular and B(M_{2} × M_{2}) ⊂ 𝔪. We already know that M is degenerate, so dim M_{2} > 0. Since dim M = 2 and dim M_{1} is even, we must have M_{1} = 0. Hence \overline{M} is quasilinear. But M is not quasilinear, since E is not quasilinear. Therefore we have a representation

M ≅ [{a \atop t^{n}c} {t^{n}c \atop b}]

with n ∈ ℕ and units a, b, c ∈ 𝔬^{∗}. The space \overline{M} ≅ [\overline{a}, \overline{b}] is anisotropic and so the element \overline{a}\overline{b} is not a square in k. Over K we have

[{1 \atop 1} {1 \atop t^{−1}} ] ≅ K ⊗ M ≅ ⟨a⟩ ⊗ [{1 \atop t^{n}ca^{−1}} {t^{n}ca^{−1} \atop ba^{−1}} ] ≅ ⟨a⟩ ⊗ [{1 \atop 1} {1 \atop abc^{−2}t^{−2n}} ] .

Comparing Arf-invariants shows

(∗)  abc^{−2}t^{−2n} = t^{−1} + x^{2} + x

with x ∈ K. Let v: = K → ℤ ∪ ∞ be the valuation associated to 𝔬. From (∗)

we get v(x) < 0 and thus v(x) = −n. Therefore x = t^{−n} ∑_{i=0}^{∞} x_{i}t^{i} with x_{i} ∈ k,

x_{0} ≠ 0. Comparing the coefficients of t^{−2n} on the left and righthand side of (∗) gives

\overline{a}\overline{b}\overline{c}^{−2} = x_{0}^{2}.

So \overline{a}\overline{b} is a square in k after all, a contradiction. Therefore E does not have FR.

We continue with the general theory. So λ: K → L ∪ ∞ is again an arbitrary place. We immediately obtain a consequence of Theorem 2 which is not contained in the results of §8 in the case of good reduction.

Corollary 2. Let E and F be spaces over K which have FR with respect to λ and with F < E. Then λ_{∗}F < λ_{∗}E.

This corollary engenders a substitution principle for quadratic forms, modeled on §3, Theorem 5.

Theorem 3. Let (g_{kl}(t))_{1≤k,l≤m} and (f_{ij}(t))_{1^{≤}i,j≤n} be symmetric matrices whose coefficients are polynomials in variables t = (t_{1}, …, t_{r}) over a field k. Let L ⊃ k be a field extension of k and c = (c_{1}, …, c_{r}) an r-tuple with coefficients in L. Suppose that the quadratic forms [g_{kl}(t)] and [f_{ij}(t)] over k(t) satisfy [g_{kl}(t)] < [f_{ij}(t)]. Suppose also that the quadratic forms [g_{kl}(c)] and [f_{ij}(c)] over L are nondegenerate. Then [g_{kl}(c)] < [f_{ij}(c)] (over L).

We return to an arbitrary place λ: K → L ∪ ∞.

Theorem 4. (Extension of §8, Lemma 2(b).) Let G be a space and F a strictly regular space over K. Suppose that F has GR with respect to λ. Finally, let E: = F ⊥ G.

Claim: E has FR with respect to λ if and only if G has FR with respect to λ. In this case we have λ_{∗}E ≅ λ_{∗}F ⊥ λ_{∗}G.

Proof. We have F = K⊗M where M is a strictly regular quadratic 𝔬-module. If G has FR with respect to λ, then G = K ⊗ P where P is a reduced nondegenerate quadratic 𝔬-module. The quadratic 𝔬-module N: = M ⊥ P is then also reduced nondegenerate and

L ⊗_{λ} N = (L ⊗_{λ} M) ⊥ (L ⊗_{λ} P) = F ⊥ G = E.

Hence E has FR with respect to λ and λ_{∗}E ≅ λ_{∗}F ⊥ λ_{∗}G.

Suppose now that E has FR with respect to λ. We have E ≅ K ⊗ N where N is a reduced nondegenerate quadratic 𝔬-module. By Theorem 1, K⊗M < K⊗N implies that N ≅ M ⊥ P where P is a reduced nondegenerate quadratic 𝔬-module. Hence

F ⊥ (K ⊗ P) ≅ E ≅ F ⊥ G.

Now G ≅ K ⊗ P by the Cancellation Theorem over K (§6, Theorem 2). Therefore G has FR with respect to λ.

To the currently developed theory of fair reduction, we can associate a variation of the weak specialization theory of §7, which we will briefly present in the following.

Theorem 5. (Extension of §6, Theorem 5.) Let 𝔬 be quadratically henselian. Let (M, q) be a reduced nondegenerate and anisotropic quadratic 𝔬-module. Furthermore, let e be a primitive vector in M. Then q(e) ∈ 𝔬^{∗}.

Proof. We choose a decomposition M = N ⊥ M′ with N strictly regular and B(M′ × M′) ⊂ 𝔪. Traversing the proof of §6, Theorem 5 and replacing M^{⊥} by M′ everywhere will give proof since all arguments will faithfully remain valid.

As before, 𝔬^{h} denotes the henselization of the valuation ring 𝔬.

Lemma 4. Let M be a quadratic 𝔬-module. Then M^{h}: = 𝔬^{h} ⊗_{𝔬} M is reduced nondegenerate if and only if M is reduced nondegenerate.

Proof. This is evident since 𝔬^{h}/𝔪^{h} = k, so that M^{h}/𝔪^{h}M^{h} is canonically isomorphic to the quadratic k-module M/𝔪M.

Now the road is clear to extend, in an appropriate way, the main result of §7 (Theorem 4), using fair reduction. As before, let λ: K → L ∪ ∞ be a place with associated valuation ring 𝔬. Let S be a system of representatives of Q(K)/Q(𝔬) in 𝔬, as introduced in §7.

Definition 4. Let E = (E, q) be a quadratic K-module (always free of finite rank). We say that E is weakly obedient with respect to λ (or: with respect to 𝔬) when E has a decomposition

(∗)  E = ⊥_{s∈S} E_{s}

such that the quadratic K-module (E_{s}, s^{−1}(q|E_{s})) has FR with respect to λ for every s ∈ S. Every decomposition of the form (∗) is called a weakly λ-modular (or: weakly 𝔬-modular) decomposition of E.

Remark. Let E be weakly obedient with respect to λ. Then

E ≅ ⊥_{s∈S} ⟨s⟩ ⊗ (K ⊗ M_{s})

with reduced nondegenerate quadratic 𝔬-modules M_{s}. For every s ∈ S we may choose a decomposition M_{s} = N_{s} ⊥ M′_{s} with strictly regular N_{s} and B_{s}(M′_{s} × M′_{s}) ⊂ 𝔪, where B_{s} is the bilinear form associated to M_{s}. Just as indicated in §7, we see that

G: = ⊥_{s∈S} ⟨s⟩ ⊗ (K ⊗ M′_{s})

is anisotropic. Furthermore,

F: = ⊥_{s∈S} ⟨s⟩ ⊗ (K ⊗ N_{s})

is strictly regular. We have E ≅ F ⊥ G. Therefore E is definitely not degenerate and is thus a space over K.

Theorem 6. (Extension of §7, Theorem 4.) Let E be a space over K which is weakly obedient with respect to λ and let

E = ⊥_{s∈S} E_{s} = ⊥_{s∈S} F_{s}

be two weakly λ-modular decompositions of E. Then^{18} λ_{∗}(E_{1}) ∼ λ_{∗}(F_{1}).

Proof. The arguments in the proof of §7, Theorem 4, remain valid in the current more general situation. One should use Theorem 5 and Lemma 4 above.

Now the following definition makes sense:

Definition 5. Let E be a quadratic space, weakly obedient with respect to λ and let E = ⊥_{s∈S} E_{s} be a weakly λ-modular decomposition of E. Then we

call λ_{∗}(E_{1}) a weak specialization of E with respect to λ. We denote the Witt class of λ_{∗}(E_{1}) by λ_{W} (E). In other words, λ_{W} (E): = {λ_{∗}(E_{1})} ∈ Ŵq(L). By Theorem 6, λ_{W} (E) is uniquely determined by E and λ.

Remark. Our proof of Theorem 6 is independent of the main result Theorem 2 of the specialization theory developed above. We could also have deduced the important Corollary 1 of Theorem 2 (specialization by FR is well-defined) from Theorem 6, analogous to the proof of §8, Theorem 1 in the quadratic case. In other words, we could have established the weak specialization theory first and then develop from this the basic idea of specialization by FR (Definition 3 above), just as before in §7 and §8.

In the theorems of §8 about the weak specialization of tensor products (Theorem 6(ii), Theorem 7) we may not simply replace the word GR by FR and the word “obedient” by “weakly obedient”. This already shows us that in our theory, good reduction does not become superfluous in any way after the introduction of fair reduction.

More important even is the observation that in the theory of generic splitting in §9 and §10 good reduction appears center stage, and not fair reduction: For example, let φ be a regular quadratic form over a field k and let L ⊃ k be a field extension. Let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower of φ. If λ: K_{r} → L ∪ ∞ is a place over k for some r ∈ {1, …, h}, then the kernel form φ_{r} of φ ⊗ K_{r} automatically has good reduction with respect to λ.

18See §6, Definition 10, for the definition of Witt equivalence ∼.

92 Chapter II. Generic Splitting Theory

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§12 Unified Theory of Generic Splitting

Just as in §9 and §10 we will not use geometric language and talk of quadratic forms instead of quadratic spaces in this section. The definitions concerning quadratic spaces over fields, encountered in §11, will be faithfully adopted in the language of forms. In what follows, a “form” will always be understood to be a quadratic form.

If γ: k → L ∪ ∞ is a place and φ a regular form over k which has GR with respect to γ, then the splitting behaviour of γ_{∗}(φ) under extensions of the field L is controlled by a given generic splitting tower (K_{r} | 0 ≤ r ≤ h) of φ, as seen in §4 and §9 (§9, Theorem 4 and generalization of §4, Scholium 3). According to §10, something similar happens for a place γ and a form φ over k having GR with respect to γ for which γ_{∗}(φ) is nondegenerate, provided some care is exercised. Now we want to extend these results to the case where φ has just FR with respect to γ.

At the same time we want to unite the results of §9 and §10 under one roof, starting with the following definition.

Definition 1. Let φ be a nondegenerate form over a field K. We call a field extension K ↪ L φ-conservative (or: conservative for φ) if the form φ ⊗ L is again nondegenerate, i.e. if the quasilinear part QL(φ) of φ remains anisotropic under a field extension from K to L.

If φ is regular, then dim QL(φ) ≤ 1 and so every field extension of K is φconservative. This is true in particular when char K ≠ 2. If char K = 2, then every separable field extension K ↪ L is φ-conservative by §10, Theorem 1.

Theorem 1. Let λ: K → L ∪ ∞ be a place, K′ ⊃ K a field extension and μ: K′ → L ∪ ∞ an extension of λ. Let φ be a (nondegenerate) form over K, having FR (resp. GR) with respect to λ, and let λ_{∗}(φ) be nondegenerate.

Claim: the extension K ↪ K′ is φ-conservative. The form φ ⊗ K′ has FR (resp. GR) with respect to μ and μ_{∗}(φ ⊗ K′) ≅ λ_{∗}(φ).

Proof. In the case of good reduction, this is Theorem 2 from §10. The proof in the case of fair reduction is similar and goes as follows.

Let 𝔬: = 𝔬_{λ} and 𝔬′: = 𝔬_{μ}. Let E = (E, q) be a space corresponding to φ and M a reduced nondegenerate quadratic 𝔬-module with E ≅ K ⊗_{𝔬}M. Then λ_{∗}(φ) corresponds to the space λ_{∗}(E) ≅ L ⊗\overline{λ} \overline{M}, where \overline{λ}: κ(𝔬) ↪ L is the field extension associated to λ. We have the factorization \overline{λ} = \overline{μ} ∘ j, featuring the inclusion j: κ(𝔬) ↪ κ(𝔬′) and the field homomorphism \overline{μ}: κ(𝔬′) ↪ L, determined by μ. Let \overline{φ} be the form over κ(𝔬), associated to \overline{M}, i.e. λ_{∗}(φ) = \overline{φ} ⊗\overline{λ} L. Since λ_{∗}(φ) is nondegenerate by assumption, \overline{λ}: κ(𝔬) ↪ L is conservative for \overline{φ}. Therefore j: κ(𝔬) ↪ κ(𝔬′) is conservative for \overline{φ} and \overline{μ}: κ(𝔬′) ↪ L is conservative for \overline{φ} ⊗ κ(𝔬′). Now \overline{φ} ⊗ κ(𝔬′) belongs to the quadratic κ(𝔬′)- module \overline{M′} with M′: = 𝔬′ ⊗_{𝔬} M. Therefore \overline{M′} is nondegenerate, in other words M′ is reduced nondegenerate. We have \overline{M′} = κ(𝔬′) ⊗_{κ(𝔬)} \overline{M}. Furthermore, K′ ⊗ E ≅ K′ ⊗_{𝔬′} M′. Thus φ ⊗ K′ has FR with respect to μ and

μ_{∗}(φ ⊗ K′) ≅ (\overline{φ} ⊗ κ(𝔬′)) ⊗\overline{μ} L = \overline{φ} ⊗\overline{λ} L ≅ λ_{∗}(φ).

In particular, φ ⊗ K′ is nondegenerate (cf. §11, the Remark following Definition 3). We conclude that K ↪ K′ is φ-conservative.

In the following let φ be a nondegenerate form over a field k. Over every field K we denote the form [{0 \atop 1} {1 \atop 0}] by H.

Theorem 2. Let K and L be extensions of the field k and let λ: K → L ∪ ∞ be a place over k. Suppose further that L is φ-conservative and let φ ⊗ K ≅ r × H ⊥ ψ be the Witt decomposition of φ.

Claim: K is also a φ-conservative extension of k. The form ψ has GR with respect to λ, and φ⊗L ≅ r×H ⊥ λ_{∗}(ψ). Thus ind(φ⊗L) ≥ ind(φ⊗K).

Proof. We apply Theorem 1 to the trivial place γ: k ↪ L and its extension λ. Now φ has GR with respect to γ and γ_{∗}(φ) = φ ⊗ L. By Theorem 1, K is conservative for φ, in other words φ ⊗ K is nondegenerate. Again by this theorem, φ ⊗ K has GR with respect to λ and λ_{∗}(φ ⊗ K) ≅ φ ⊗ L. Since r × H also has GR with respect to λ, it follows that ψ has GR with respect to λ (§8, Theorem 2), and we have φ ⊗ L ≅ r × H ⊥ λ_{∗}(ψ).

Corollary. Let K and L be specialization equivalent extensions of the field k. Then K is φ-conservative if and only if L is φ-conservative. In this case we have ind(φ ⊗ K) = ind(φ ⊗ L). Furthermore, if λ: K → L ∪ ∞ is a place over k, then ker(φ⊗K) has GR with respect to λ and λ_{∗}(ker(φ⊗K)) ≅ ker(φ⊗L).

Next we define a generic splitting tower (K_{r} | 0 ≤ r ≤ h) associated to φ just as in §4 for char k ≠ 2 and in §9 for regular forms, with higher kernel forms φ_{r} and higher indices i_{r} (0 ≤ r ≤ h): K_{0} is an inessential extension of k, φ_{r}: = ker(φ ⊗ K_{r}), K_{r+1} ∼_{k} K_{r}(φ_{r}), i_{r+1}: = ind(φ_{r} ⊗ K_{r+1}), in case r < h. The construction stops with step h if dim φ_{h} − dim QL(φ_{h}) ≤ 1.

All this makes sense, since an inductive argument shows that QL(φ_{r}) = QL(φ) ⊗ K_{r} is anisotropic for every r ∈ {0, … h}. Indeed, the field extension K_{r}(φ_{r})/K_{r} is separable for r < h. Hence we can conclude by Theorem 2 that K_{r+1}/K_{r} is φ_{r}-conservative once we already know that QL(φ_{r}) is anisotropic, i.e. that φ_{r} is nondegenerate. The extension K_{h}/k is therefore φ-conservative.

Note. If char k ≠ 2, then dim φ_{h} ≤ 1. If char k = 2, then φ_{h} = QL(φ_{h}).

We now have to convince ourselves that the field tower (K_{r} | 0 ≤ r ≤ h) really accomplishes what we expect from a generic splitting tower. As before in more special situations, it suffices for this purpose, to prove a theorem of the following sort.

Theorem 3 (Extension of §9, Theorem 4 and §10, Theorem 3). Let γ: k → L ∪ ∞ be a place. Suppose that the form φ has FR with respect to γ and that γ_{∗}(φ) is nondegenerate. Then γ_{∗}(φ) is isotropic if and only if γ can be extended to a place λ: k(φ) → L ∪ ∞.

Proof. We have φ ⊗ k(φ) ≅ H ⊥ ψ for a form ψ over k(φ). If there exists a place λ: k(φ) → L ∪ ∞ then, by Theorem 1, φ ⊗ k(φ) has FR with respect to λ and λ_{∗}(φ ⊗ k(φ)) ≅ γ_{∗}(φ). Then it follows from §11, Theorem 4, that ψ has FR with respect to λ and that γ_{∗}(φ) ≅ H ⊥ λ_{∗}(ψ). Hence γ_{∗}(φ) is isotropic.

Conversely, suppose that the nondegenerate form \overline{φ}: = γ_{∗}(φ) is isotropic. Since H < \overline{φ} we have dim \overline{φ} − dim QL(\overline{φ}) ≥ 2. Now, by earlier work, we see that γ can be extended to a place γ̃: k(φ) → L(\overline{φ}) ∪ ∞ (look again at the proof of §9, Theorem 3). \{Note: here we do not need the assumption that \overline{φ} is isotropic, but only that dim \overline{φ}−dim QL(\overline{φ}) ≥ 2.\} Since \overline{φ} is isotropic, L(\overline{φ}) is a purely transcendental extension of L (§9, Lemma 1). Thus there exists a place ρ: L(\overline{φ}) → L ∪ ∞ over L. The place λ: = ρ ∘ γ̃: k(φ) → L ∪ ∞ is an extension of γ.

In what follows, let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower associated to φ with higher kernel forms φ_{r} and indices i_{r}.

Theorem 4 (Extension of §4, Theorem 3 and §10, Theorem 4). Let γ: k → L∪∞ be a place with respect to which φ has FR (resp. GR). Suppose that the form γ_{∗}(φ) is nondegenerate. Finally, suppose that λ: K_{m} → L∪∞ is a place for some m ∈ {0, …, h}, which extends γ and which cannot be extended to K_{m+1} in case m < h. Then φ_{m} has FR (resp. GR) with respect to λ. The form γ_{∗}(φ) has kernel form λ_{∗}(φ_{m}) and Witt index i_{0} + … + i_{m}.

Proof. We give the proof for FR. The case of GR can be treated in an analogous way. We have an isometry

(1)  φ ⊗ K_{m} ≅ φ_{m} ⊥ (i_{0} + … + i_{m}) × H.

By Theorem 1, φ ⊗ K_{m} has FR with respect to λ and λ_{∗}(φ ⊗ K_{m}) ≅ γ_{∗}(φ). From (1) we get, according to §11, Theorem 4, that φ_{m} has FR with respect to λ and that

(2)  γ_{∗}(φ) ≅ λ_{∗}(φ_{m}) ⊥ (i_{0} + … + i_{m}) × H.

In particular, λ_{∗}(φ_{m}) is nondegenerate. If λ_{∗}(φ_{m}) were isotropic, then we would surely have that dim φ_{m} ≥ dim QL(φ_{m}) + 2, i.e. m < h. But then it follows from Theorem 3 that λ can be extended to a place μ: K_{m+1} → L ∪ ∞, contradiction! Therefore λ_{∗}(φ_{m}) is anisotropic and (2) is the Witt decomposition of γ_{∗}(φ_{m}).

By applying this theorem to the trivial place γ, we obtain

Scholium 1 (Extension of §4, Scholium 1). Let L ⊃ k be a φ-conservative field extension.

(1) Let λ: K_{m} → L ∪ ∞ be a place over k for some m ∈ {0, …, h}, which cannot be extended to a place from K_{m+1} to L in case m < h. Then φ_{m} has good reduction with respect to λ and λ_{∗}(φ_{m}) is the kernel form of φ ⊗ L.

(2) If λ′: K_{r} → L ∪ ∞ is a place over k, then r ≤ m and λ′ can be extended to a place μ: K_{m} → L ∪ ∞.

(3) For a given number t with

0 ≤ t ≤ [\frac{dim φ − dim QL(φ)}{2}] = i_{0} + … + i_{h}

and m ∈ ℕ_{0} minimal such that t ≤ i_{0} + … + i_{m}, K_{m} is a generic φ-conservative field extension of k for the splitting off of t hyperbolic planes of φ (cf. the problem posed at the beginning of §4).

Thus, the field tower (K_{r} | 0 ≤ r ≤ h) rightfully merits the name “generic splitting tower for φ” also in the current general situation.

Definition 2. As before we call h the height of the form φ and write h = h(φ). Further, we define the splitting pattern SP(φ) as the set of all indices ind(φ ⊗ L), where L runs through all φ-conservative field extensions of k. By the scholium we have

SP(φ) = {i_{0} + … + i_{r} | 0 ≤ r ≤ h}.

Definition 3. We call any field extension E ⊃ k which is specialization equivalent to k(φ) over k a generic zero field of φ. Further, for r ∈ {0, …, h}, we call any field extension F ⊃ k which is specialization equivalent to K_{r} over k a partial generic splitting field of φ or, more precisely, a generic splitting field of level r. This signifies that F is generic for the splitting off of as many hyperbolic planes as the (r + 1)-th number i_{0} + … + i_{r} in SP(φ) indicates. In case r = h, we speak of a total generic splitting field.

This terminology generalizes notions from §4 and §9, but one has to exercise some caution; the extensions E ⊃ k and F ⊃ k have the mentioned generic properties – as far as we know – only within the class of φ-conservative field extensions of k.

Comment on the Theory so far. If the form φ over k is degenerate, but dim φ − dim QL(φ) ≥ 2, then the function field k(φ) can be constructed just as in the nondegenerate case. Thus we can formally define a “generic splitting tower” (K_{r} | 0 ≤ r ≤ h) as above. We may also inquire about the splitting behaviour of φ under extensions L ⊃ k of the field k, in other words about the possibilities for the index ind(φ ⊗ L) and the kernel form ker(φ ⊗ L) (see Definition 10 in §6). However, can we also extend the theory so far to degenerate forms?

We have a decomposition φ ≅ φ̂ ⊥ δ(φ) with δ(φ) ≅ t × [0] and φ̂ nondegenerate, cf. §6, Definition 9 ff. Clearly k(φ) is a purely transcendental extension of k(φ̂) of transcendence degree t. Therefore k(φ) ∼_{k} k(φ̂). Consequently, the above tower (K_{r} | 0 ≤ r ≤ h) is a generic splitting tower of φ̂. It is now clear that our theorems so far all remain valid for degenerate φ, as long as we complete Definition 1 above as follows: for degenerate φ, a field extension K ↪ L is called conservative for φ if it is conservative for φ̂.

This, however, furnishes us with a fairly bland extension of the theory so far to degenerate forms. The obvious, difficult question seems to be: Let φ be a nondegenerate form over a field k and L ⊃ k a field extension with φ ⊗ L degenerate. Is it so that the splitting behaviour of φ⊗L under field extensions of L is controlled by a given generic splitting tower (K_{r} | 0 ≤ r ≤ h) of φ in a similar way as indicated in Scholium 1(1) above? For example, is SP(φ ⊗ L) ⊂ SP(φ)?

Our theory does not give any information here. The main problem seems to occur in Theorem 1 above. If we do not know there that λ_{∗}(φ) is nondegenerate, we can not conclude – as far as I can see – that (φ ⊗ K′)^{∧} has fair reduction with respect to μ.

After this digression, we suppose again that φ is a nondegenerate form over k and that (K_{r} | 0 ≤ r ≤ h) is a generic splitting tower of φ with higher kernel forms φ_{r} and indices i_{r}. From Theorem 4 above we immediately obtain a literal repetition of §4, Scholium 2 in the current, more general situation, if we bear in mind that for every generic splitting tower (K′_{s} | 0 ≤ s ≤ h) of φ, all extensions K′_{s} of k are conservative for φ. Furthermore, we obtain from Theorem 4 an extension of §4, Scholium 3 in the same way as we obtained §4, Scholium 3 from §4, Theorem 3:

Scholium 2. Let γ: k → L ∪ ∞ be a place with respect to which φ has FR (resp. GR). Suppose that γ_{∗}(φ) is nondegenerate. Then:

(1) SP(γ_{∗}(φ)) ⊂ SP(φ).

(2) The higher kernel forms of γ_{∗}(φ) arise from certain higher kernel forms of φ by means of specialization. More precisely: if (L_{s} | 0 ≤ s ≤ e) is a generic splitting tower of γ_{∗}(φ), then e ≤ h and, for every s with 0 ≤ s ≤ e, we have

ind(γ_{∗}(φ) ⊗ L_{s}) = i_{0} + … + i_{m}

with m ∈ {0, …, h}. The number m is the biggest integer such that γ can be extended to λ: K_{m} → L_{s} ∪ ∞. The kernel form φ_{m} of φ ⊗ K_{m} has FR (resp. GR) with respect to every extension λ of this kind, and λ_{∗}(φ_{m}) is the kernel form of γ_{∗}(φ) ⊗ L_{s}.

(3) If ρ: K_{r} → L_{s} ∪ ∞ is a place, which extends γ: k → L ∪ ∞, then r ≤ m and ρ can be further extended to a place from K_{m} to L_{s}.

§13 Regular Generic Splitting Towers and Base Extension

Before turning towards generic splitting towers, we will give two general definitions which will also serve us well in later sections.

As before, the word form over a field k will always be understood to mean a nondegenerate quadratic form over k. In the following let φ be a form over k.

Definition 1. Let dim φ be even, φ ≠ 0 and QL(φ) = 0.

(a) The discriminant algebra Δ(φ) is defined as follows: If char k ≠ 2, we let Δ(φ): = k[X]/(X^{2} − a), where a is a representative of the signed determinant d(φ̃) = ak^{∗2} of the bilinear form φ̃ = B_{φ} associated to φ (cf. §2). If char k = 2, we let Δ(φ): = k[X]/(X^{2} + X + c), where c is a representative of the Arf-invariant Arf(φ) ∈ k^{+}/℘k.

(b) We define the discriminant of φ to be the isomorphism class of Δ(φ) as k-algebra. We will denote it sloppily by Δ(φ) as well. The discriminant is independent of the choice of a resp. c above.

(c) We say that Δ(φ) splits when Δ(φ) is not a field, i.e. when Δ(φ) ≅ k ×k. We will symbolically write Δ(φ) = 1 when φ splits and Δ(φ) ≠ 1 when φ doesn’t split.

Remark 1. This notation is not completely groundless, for the isomorphism classes of quadratic separable k-algebras form a group in a natural way with unit element k × k. In this group the equality Δ(φ⊥ψ) = Δ(φ)Δ(ψ) holds. We will not discuss the group of quadratic separable k-algebras any deeper since we will not make any serious use of it.

Remark 2. If K/k is a field extension,^{19} then Δ(φ ⊗ K) = Δ(φ) ⊗_{k} K.

Remark 3. If τ is the norm form of the k-algebra Δ(φ), then Δ(φ) = Δ(τ). Is further dim φ = 2, then φ ≅ cτ for some c ∈ k^{∗}. If we write τ = [{1 \atop 1} {1 \atop δ}], then Δ(φ) = k[X]/(X^{2} + X + δ), also when char k ≠ 2.

Definition 2 (cf. [KR, Def.1.1]). Let dim φ > 1. We say that “φ is of outer type” when dim φ is even, QL(φ) = 0 and Δ(φ) ≠ 1. In all other cases (thus in particular when QL(φ) ≠ 0) we say that “φ is of inner type”.

19From now on we will often denote a field extension F ↪ E by E/F, as has been customary in algebra for a long time.

Remark 4. For those readers who are at home in the theory of reductive algebraic groups, we want to remark that Definition 2 leans on the concepts of inner/outer type in use there: is φ regular (i.e. dim QL(φ) ≤ 1) and dim φ ≥ 3, then the group SO(φ) is almost simple. This group is of inner/outer type if and only if this is the case for φ in the sense of Definition 2.

We want to construct “regular” generic splitting towers of φ, having particularly convenient properties with respect to extensions of the ground field k, which are nonetheless sufficiently general. Starting with such a regular generic splitting tower of φ we will then construct a regular generic splitting tower of φ ⊗ L for every φ-conservative field extension L/k.

For every form ψ with dim ψ ≥ 2+dim QL(ψ) over a field K we introduced the field extension K(ψ) of K earlier (§4; §9; §10, following Def.2). K(ψ) was defined to be the function field of the affine quadric ψ(X) = 0 over K, except when ψ ≅ [{0 \atop 1} {1 \atop 0}], in which case this quadric degenerates in two lines. In this situation we defined K(ψ) = K(t) for some indeterminate t over K. Sometimes it is more natural to use the projective quadric ψ(X) = 0 over K instead of the affine quadric ψ(X) = 0. Thus we arrive at the following

Definition 3. Let ψ be a (nondegenerate quadratic) form over K with dim ψ ≥ 2 + dim QL(ψ). We define a field extension K{ψ} of K as follows: If ψ ≅ [{0 \atop 1} {1 \atop 0}], we let K\{ψ\} = K. Otherwise we let K\{ψ\} be the subfield K(\frac{x_{1}}{x_{i}} , …, \frac{x_{n}}{x_{i}}) of K(ψ) = QuotK[X_{1}, …, X_{n}]/ψ(X_{1}, …, X_{n}) = K(x_{1}, …, x_{n}) for some i ∈ [1, n].^{20}

In the main case ψ ≅ [{0 \atop 1} {1 \atop 0}] this field is obviously independent of the choice of i ∈ [1, n]. We have K(ψ) = K{ψ}(x_{i}) and x_{i} is transcendental over K\{ψ\}. Also in the case ψ ≅ [{0 \atop 1} {1 \atop 0}] we have K(ψ) = K\{ψ\}(t) where t is an indeterminate.

K(ψ) is an inessential extension of K{ψ}. Thus, all this time we could have used K{ψ} instead of K(ψ).

Back to our form φ over k! Let h = h(φ) be the height of φ.

Definition 4. The projective standard tower of φ is the field tower (K_{r} | 0 ≤ r ≤ h) with K_{0} = k, K_{r+1} = K_{r}{φ_{r}} (0 ≤ r ≤ h − 1), where φ_{r} denotes the kernel form of φ ⊗ K_{r}. Similarly one obtains the affine standard tower (K′_{r} | 0 ≤ r ≤ h) of φ by replacing the K_{r}{ψ_{r}} with the function fields K_{r}(φ_{r}) of the affine quadrics φ_{r} = 0.

Remark 5. From the definition of generic splitting tower (§12, just after Theorem 2) it follows immediately that (K_{r} | 0 ≤ r ≤ h) and (K′_{r} | 0 ≤ r ≤ h) are both generic splitting towers of φ. For every r ∈ {0, …, h} we have that K′_{r} is a purely transcendental field extension of K_{r} of transcendence degree r.

20In [K_{5}] this field is denoted by K(ψ)_{0}. [1, n] denotes the set {1, 2, …, n}.

§13 Regular Generic Splitting Towers and Base Extension 101

Let us recall that a field extension L/K is called regular when it is separable and K is algebraically closed in L. This is synonymous with saying that L and the algebraic closure K̃ of K in L are linearly disjoint, i.e. that L ⊗_{K} K̃ is a field, cf. [Lg_{2}, Chap. 3]. If L/K is regular, then the K-algebra L ⊗_{K} E does not contain zero divisors for any field extension E/K.

Theorem 1. Let (K_{r}|0 ≤ r ≤ h) be the projective standard tower of φ. Then the field extensions K_{r}/K_{r−1} with 0 < r < h are all regular. The same is true for K_{h}/K_{h−1} in case h > 0 and φ is of inner type. Furthermore, the kernel form φ_{h−1} of φ ⊗ K_{h−1} has dimension ≥ 3 in this situation. If φ is of outer type, then h ≥ 1 and K_{h} = K_{h−1} ⊗_{k} Δ(φ). Furthermore, we then have φ_{h−1} ≅ c(τ ⊗ K_{h−1}), where τ is the norm form of the field extension Δ(φ) of k and c ∈ K_{h−1}^{∗} is a constant. In particular we have dim φ_{h−1} = 2.

Proof. If ψ is a nondegenerate quadratic form over a field K and if dim ψ > 2, then the extension K{ψ}/K is clearly regular. This established the first statement of the theorem. If h > 0 and φ is of inner type, then φ ⊗ K_{h−1} is also of inner type and thus the kernel form φ_{h−1} of φ⊗K_{h−1} is of inner type. This shows that dim φ_{h−1} ≥ 3 and so K_{h} = K_{h−1}{φ_{h−1}} over K_{h−1} is again regular. Suppose now that φ is of outer type. φ does not split, so that h > 0. The extension K_{h−1}/K_{0} is regular and K_{0}/k is inessential. Therefore k is algebraically closed in K_{h−1} and Δ(φ_{h−1}) = Δ(φ ⊗ K_{h−1}) = Δ(φ) ⊗_{k} K_{h−1} is a field. If it would be true that dim φ_{h−1} > 2, then K_{h} = K_{h−1}{φ_{h−1}} over K_{h−1} would be regular. However, Δ(φ_{h−1}) ⊗_{K_{h−1}} K_{h} = Δ(φ ⊗ K_{h}) is not a field since φ ⊗ K_{h} ∼ 0. Therefore the dimension of φ_{h−1} is 2. By Remarks 2 and 3 above, we have Δ(φ_{h−1}) = Δ(φ⊗K_{h−1}) = Δ(φ)⊗_{k} K_{h−1} = Δ(τ)⊗_{k} K_{h−1} = Δ(τ ⊗K_{h−1}) (and τ ⊗K_{h−1} is the norm form of K_{h}/K_{h−1}). We conclude that φ_{h−1} ≅ c(τ ⊗ K_{h−1}) for some element c of K_{h−1}.

Theorem 2. Let r ∈ [0, h] and let E/k be a partially generic splitting field of level r of φ.

(a) If r < h, then k is algebraically closed in E.

(b) If r = h and φ is of inner type, then k is likewise algebraically closed in E.

(c) If r = h and φ is of outer type, then k has algebraic closure Δ(φ) in E.

Proof. By Theorem 1 these statements hold when E is the r-field K_{r} in the projective standard tower (K_{i} | 0 ≤ i ≤ h) of φ. The theorem now follows in all generality from the following simple lemma.

Lemma 1. Let K and L be extensions of the field k which are specialization equivalent, K ∼_{k} L. Then every place λ: K → L∪∞ over k maps the algebraic closure of k in K isomorphically to the algebraic closure of k in L.

Proof of the lemma. Let K^{∘} be the algebraic closure of k in K and L^{∘} the algebraic closure of k in L. Let λ: K → L ∪ ∞ and μ: L → K ∪ ∞ be places over k. Since K^{∘} and L^{∘} are algebraic over k, the places λ and μ are finite on K^{∘} resp. L^{∘}. Their restrictions to K^{∘} and L^{∘} are thus field homomorphisms λ′: K^{∘} → L^{∘}, μ′: L^{∘} → K^{∘}, which are both the identity on k. Now μ′ ∘λ′ is an endomorphism of K^{∘}/k and thus automatically an automorphism of K^{∘}/k. Likewise, λ′ ∘ μ′ is an automorphism of L^{∘}/k. The statement of the lemma is now clear.

Now we can precisely formulate a desirable property of generic splitting towers.

Definition 5. We call a generic splitting tower (K_{r} | 0 ≤ r ≤ h) of φ regular when the following holds:

(1) For every r with 0 ≤ r < h − 1 the extension K_{r+1}/K_{r} is regular.

(2) If φ is of inner type and h > 0, then K_{h}/K_{h−1} is also regular. On the other hand, if φ is of outer type, then K_{h} is regular over the composite K_{h−1} · Δ(φ) = K_{h−1} ⊗_{k} Δ(φ).

Examples. The projective standard tower of φ is regular by Theorem 1. The affine standard tower of φ is likewise regular.

Now let (K_{r} | 0 ≤ r ≤ h) be a regular generic splitting tower of φ and let L/k be a φ-conservative field extension of k. Using (K_{r} | 0 ≤ r ≤ h) and L we want to construct a generic splitting tower of φ ⊗ L.

Definition 6. For every r ∈ {0, 1, …, h} we construct a field composite K_{r} · L of K_{r} and L over k as follows: if r < h, or if r = h and φ is of inner type, then K_{r} ⊗_{k} L is free of zero divisors and we let K_{r} · L be the quotient field of this ring, i.e. the uniquely determined free composite of K and L over k. If r = h and φ is of outer type, we distinguish the cases where φ ⊗ L is of outer/inner type.

First, let φ ⊗ L be of outer type. Then Δ(φ) ⊗_{k} L is a field. Let K_{h} · L be the free composite of K_{h} with Δ(φ) ⊗_{k} L over Δ(φ), i.e. again the quotient field of K_{h} ⊗_{Δ(φ)} (Δ(φ) ⊗_{k} L) = K_{h} ⊗_{k} L. Finally, let φ ⊗ L be of inner type. Now Δ(φ) can be embedded over k in L in two ways. We choose one such embedding and set K_{h} · L = Quot(K_{h} ⊗_{Δ(φ)} L).

If necessary, we write more precisely K_{r} ·_{k} L (0 ≤ r ≤ h) for the field composite K_{r} · L. Later on we will call K_{r} · L sloppily “the” free composite of K_{r} and L over k, also in the case r = h, φ of outer type, φ ⊗ L of inner type. It will never matter which of the two embeddings Δ(φ) ↪ L we have chosen.

We use φ_{r} to denote – as before – the kernel form of φ ⊗ K_{r} (r-th higher kernel form) and i_{r} to denote the Witt index of φ ⊗ K_{r} (r-th higher index, 0 ≤ r ≤ h).

Theorem 3. Let J be the set of all r ∈ [0, h] such that φ_{r} ⊗ K_{r} · L is anisotropic, i.e. such that ind(φ⊗K_{r}) = ind(φ⊗K_{r}·L). Let r_{0} < r_{1} < ⋯ < r_{e} be the elements of J.

Claim:

(a) (K_{r_{i}} · L | 0 ≤ i ≤ e) is a regular generic splitting tower of φ ⊗ L.

(b) K_{r+1} · L/K_{r} · L is a regular inessential extension for every r ∈ [0, h] \ J.

For the proof of this technically very important theorem, we need a general lemma about places.

Lemma 2. Let K ⊃ k and L ⊃ k be arbitrary extensions of a field k and let λ: K → L ∪ ∞ be a place over k. Furthermore, let E ⊃ k be a field extension which is linearly disjoint from K and L over k. Consider the free composites K · E and L · E of K resp. L with E over k.

Claim: λ has a unique extension λ̃: K · E → L · E ∪ ∞ to a place over E.

Proof. K · E is the quotient field of K ⊗_{k} E and L · E is the quotient field of L ⊗_{k} E. Let 𝔬: = 𝔬_{λ} and let α: 𝔬 ⊗_{k} E → L ⊗_{k} E be the homomorphism of E-algebras induced by λ|𝔬: 𝔬 → L. Following the general extension theorem for places [Bo_{2}, §2, Prop.3], we choose a place μ: K · E → (L̃ · E) ∪ ∞ in the algebraic closure of L · E which extends the homomorphism α. We will now show that μ is the only such extension of α and that μ(K · E) ⊂ (L · E) ∪ ∞.

For this purpose we choose a basis (ω_{i} | i ∈ I) of E over k (as k-vector space). Let z ≠ 0 be an element in K · E. We write z = \frac{x}{y}, where x, y ∈ 𝔬 ⊗ E are both non-zero. We then have equations

x = u · ∑_{i∈I} a_{i} ⊗ ω_{i} , y = v · ∑_{i∈I} b_{i} ⊗ ω_{i}

with u, v ∈ K and families (a_{i} | i ∈ I) and (b_{i} | i ∈ I) in 𝔬, both of them not fully contained in the maximal ideal 𝔪 of 𝔬. Then

μ (∑_{i∈I} a_{i} ⊗ ω_{i}) = ∑_{i∈I} λ(a_{i})ω_{i} ≠ 0, μ (∑_{i∈I} b_{i} ⊗ ω_{i}) = ∑_{i∈I} λ(b_{i})ω_{i} ≠ 0,

since (ω_{i} | i ∈ I) is also a family of elements of L · E, linearly independent over L. We thus obtain

μ(z) = λ (\frac{u}{v}) · (∑_{i∈I} λ(a_{i})ω_{i}) · (∑_{i∈I} λ(b_{i})ω_{i})^{−1} ∈ (L · E) ∪ ∞.

Proof of Theorem 3. We assume without loss of generality that φ is anisotropic and use induction on the height h. Remark that K_{0} ∼_{k} k implies that K_{0} · L ∼_{L} L by the lemma. Therefore the extension K_{0} · L/L is inessential.

For h = 0 nothing more has to be done, so suppose that h > 0. First we assume that dim φ > 2. We will deal with the (easier) case dim φ = 2 at the end.

The form φ_{1} = ker(φ ⊗ K_{1}) has height h − 1 and regular generic splitting tower (K_{r}|1 ≤ r ≤ h). We want to apply the induction hypothesis to φ_{1}, this tower and the field extension K_{1} · L/K_{1}.

Clearly the extension K_{1}·L/K_{0}·L is regular (cf. e.g. [Lg_{2}, p.58]). Furthermore, K_{0} · L/L is inessential and L/k φ-conservative. It follows that K_{1} · L/k is φ-conservative and then that K_{1} · L/K_{1} is φ_{1}-conservative. For every r with 1 ≤ r ≤ h − 1 we have

K_{r} ·_{k} L = Quot(K_{r} ⊗_{k} L) = Quot(K_{r} ⊗_{K_{1}} (K_{1} ⊗_{k} L)) = K_{r} ·_{K_{1}} (K_{1} ·_{k} L).

Also, K_{h} ·_{k} L = K_{h} ·_{K_{1}} (K_{1} ·_{k} L). This can be seen using the same calculation in case φ is of inner type, and also when φ is of outer type and Δ(φ) cannot be embedded in L. If φ is of outer type and Δ(φ) ⊂ L, we perform the following calculation (bearing Definition 6 in mind).

K_{h} ·_{k} L: = K_{h} ·_{Δ(φ)} L = Quot(K_{h} ⊗_{Δ(φ)} L)
= Quot(K_{h} ⊗_{K_{1}·Δ(φ)} (K_{1} · Δ(φ) ⊗_{Δ(φ)} L)).

We have K_{1} · Δ(φ) = K_{1} ⊗_{k} Δ(φ) = Δ(φ_{1}) and do indeed obtain again

K_{h} ·_{k} L = Quot(K_{h} ⊗_{Δ(φ_{1})} (K_{1} ⊗_{k} L)) = K_{h} ·_{K_{1}} (K_{1} ·_{k} L).

Thus we can apply the induction hypothesis, which tells us among other things that K_{r+1} · L/K_{r} · L is regular and inessential for every r ∈ [1, h] \ J. We now distinguish the cases r_{0} ≥ 1 and r_{0} = 0.

Suppose that r_{0} ≥ 1. By the induction hypothesis, (K_{r_{i}} · L|0 ≤ i ≤ e) is a regular generic splitting tower of φ ⊗ K_{1} · L. Let F_{0}: = K_{0} · L. The form φ_{0} ⊗ F_{0} = φ ⊗ F_{0} is isotropic. Therefore the extension F_{0}(φ_{0} ⊗ F_{0}) = K_{0}(φ_{0}) ·_{K_{0}} F_{0} of F_{0} is purely transcendental, and thus inessential. By the lemma, K_{1} ·_{K_{0}} F_{0} ∼_{F_{0}} K_{0}(φ_{0}) ·_{K_{0}} F_{0}. Hence K_{1} ·_{K_{0}} F_{0}/F_{0} is also inessential. In analogy with the first calculation above, we find that

K_{1} ·_{K_{0}} F_{0} = K_{1} ·_{K_{0}} (K_{0} ·_{k} L) = K_{1} ·_{k} L,

so that K_{1}·_{k} L/K_{0}·_{k} L is inessential. This proves statement (b) for the current case.

We saw that K_{0} · L/L is inessential. Thus K_{1} · L/L is inessential. The generic splitting tower (K_{r_{i}} ·L|0 ≤ i ≤ e) of φ ⊗ K_{1}·L is thus also a generic splitting tower of φ ⊗ L.

We come to the case dim φ > 2, r_{0} = 0. Statement (b) is covered by the induction hypothesis. φ ⊗ L is anisotropic. By the induction hypothesis, φ⊗K_{1}·L has regular generic splitting tower (K_{r_{i}} ·L|1 ≤ i ≤ e). The extension K_{r_{1}} · L/K_{0} · L is regular. For the proof of statement (a), it remains to be shown that K_{r_{1}} · L is the generic zero field of φ ⊗ K_{0} · L. We already know that K_{r_{1}} ·L/K_{1} ·L is inessential. Thus it suffices to show that K_{1} · L is the generic zero field of φ ⊗ (K_{0} · L).

Let F_{0}: = K_{0} · L. We have φ_{0} ⊗ F_{0} = φ ⊗ F_{0}, and thus F_{0}(φ ⊗ F_{0}) = K_{0}(φ_{0}) ·_{K_{0}} F_{0}. Also K_{1} ∼_{K_{0}} K_{0}(φ_{0}). By the lemma, this shows that

K_{1} ·_{K_{0}} F_{0} ∼_{F_{0}} K_{0}(φ_{0}) ·_{K_{0}} F_{0}.

Furthermore, K_{1} ·_{K_{0}} F_{0} = K_{1} ·_{k} L. Thus, K_{1} ·_{k} L ∼_{F_{0}} F_{0}(φ_{0} ⊗ F_{0}), what we wanted to show.

Finally, assume that dim φ = 2. Now we have h = 1. The field Δ(φ_{0}) = Δ(φ) ⊗_{k} K_{0} is a quadratic extension of K_{0} and K_{1} is a regular extension of Δ(φ_{0}). We have a place λ: K_{1} → Δ(φ_{0}) ∪ ∞ over K_{0}. After composing λ with the non-trivial automorphism of Δ(φ_{0})/K_{0} (if necessary), we obtain a place μ: K_{1} → Δ(φ_{0}) ∪ ∞ over Δ(φ_{0}). Therefore K_{1}/Δ(φ_{0}) is an inessential extension and regular.

Let F_{0}: = K_{0} · L, F_{1}: = K_{1} · L. We again make a distinction between the cases φ ⊗ L isotropic, resp. anisotropic.

Suppose first that φ ⊗ L is isotropic. Now J = {1}, and Δ(φ) ⊗_{k} L splits. We choose an embedding Δ(φ) ↪ L and obtain from this an embedding K_{0} · Δ(φ) = Δ(φ_{0}) ↪ K_{0} · L = F_{0}. By Definition 6, we now have

K_{1}·L: = K_{1}·_{Δ(φ)}L = Quot(K_{1}⊗_{Δ(φ)}L) = Quot(K_{1}⊗_{Δ(φ_{0})}(Δ(φ_{0})⊗_{Δ(φ)}L)),

and furthermore Δ(φ_{0}) ⊗_{Δ(φ)} L = K_{0} ⊗_{k} L. It follows that

F_{1} = K_{1} · L = K_{1} ·_{Δ(φ_{0})} (K_{0} · L) = K_{1} ·_{Δ(φ_{0})} F_{0}.

It is now clear that the extension F_{1}/F_{0} is regular. Since K_{1}/Δ(φ_{0}) is inessential, it follows furthermore from the lemma that F_{1}/F_{0} is inessential. F_{0}/L is also inessential. Thus F_{1}/L is inessential. This establishes statements (a) and (b) in this case.

Finally, let dim φ = 2 and assume that φ ⊗ L is anisotropic. Now J = {0, 1}. Statement (b) is vacuous. We already know that F_{0}/L is inessential. Furthermore, K_{1}/Δ(φ_{0}) is regular and inessential. Therefore

F_{1} = K_{1} ·_{Δ(φ_{0})} (Δ(φ_{0}) ·_{K_{0}} F_{0})

is regular and inessential over Δ(φ_{0}) ·_{K_{0}} F_{0} = Δ(φ_{0} ⊗ F_{0}). Furthermore, Δ(φ_{0} ⊗ F_{0}) = F_{0}{φ_{0} ⊗ F_{0}}. Hence F_{1} is the generic zero field of the form φ_{0} ⊗ F_{0}. Therefore (F_{0}, F_{1}) is a regular generic splitting tower of φ ⊗ L.

Corollary. In the situation of Theorem 3, φ ⊗ L has height e and generic splitting pattern

SP(φ ⊗ L) = {i_{0} + … + i_{r_{j}} | 0 ≤ j ≤ e}.

For every j ∈ [0, e] is φ_{r_{j}} ⊗ (K_{r_{j}} · L) furthermore the j-th higher kernel form of φ ⊗ L with respect to the generic splitting tower constructed here.

106 Chapter II. Generic Splitting Theory

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§14 Generic Splitting Towers of a Specialized Form

In the following, let γ: k → L ∪ ∞ be a place and φ a quadratic form over k having FR with respect to γ. Assume that the form \overline{φ}: = γ_{∗}(φ) over L is nondegenerate. Finally, let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower of φ.

We have seen in §12 that the splitting behaviour of \overline{φ} with respect to \overline{φ}-conservative extensions of the field L is controlled by the tower (K_{r} | 0 ≤ r ≤ h), cf. §12, Theorem 4 and Scholium 2. Hence we may hope that it is actually possible to construct a generic splitting tower of \overline{φ} in a natural way from (K_{r} | 0 ≤ r ≤ h). We will devote ourselves to the task of constructing such a tower.

In the special case where the place γ is trivial and the tower (K_{r} | 0 ≤ r ≤ h) is regular, we already dealt with this task in the previous section (§13, Theorem 3). In general we cannot expect to find a solution as nice as the solution in §13.

Let us continue by fixing some notation. Let i_{r} be the higher indices and φ_{r} the higher kernel forms of φ with respect to the given splitting tower (K_{r} | 0 ≤ r ≤ h) of φ. Thus we have SP(φ) = {i_{0} + … + i_{r} | 0 ≤ r ≤ h}. Further, let J be the subset of \{0, 1, …, h\} with SP(\overline{φ}) = \{i_{0}+…+i_{r} | r ∈ J\}. The set J contains e: = h(\overline{φ}) elements. Let us list them as

0 ≤ t(0) < t(1) < … < t(e) = h.

We formally set t(−1) = −1.

Let (L_{s} | 0 ≤ s ≤ e) be a generic splitting tower of \overline{φ}. Following §12, Scholium 2 we choose places μ_{s}: K_{t(s)} → L_{s} ∪ ∞ (0 ≤ s ≤ e) such that μ_{0} extends the place γ and such that every μ_{s} with 1 ≤ s ≤ e extends the place μ_{s−1}. For every r ∈ {0, …, h} we have an s ∈ {0, …, e} with t(s − 1) < r ≤ t(s). Let λ_{r}: K_{r} → L_{s} ∪ ∞ be the restriction of the place μ_{s} to K_{r}. \{In particular, μ_{s} = λ_{t(s)}.\} Let ψ_{s} = ker(\overline{φ}⊗L_{s}) be the s-th higher kernel form of \overline{φ}. By §12 φ_{t(s)} has FR with respect to μ_{s} and (μ_{s})_{∗}(φ_{t(s)}) ≅ ψ_{s}. Furthermore, for t(s − 1) < r < t(s), φ_{r} also has FR with respect to λ_{r} and

(λ_{r})_{∗}(φ_{r}) ≅ ψ_{s} ⊥ (i_{r+1} + … + i_{t(s)}) × H.

Given a place α: E → F ∪ ∞ and a subfield K of E, we denote the image of K under α in general sloppily with α(K), in other words,

α(K): = α(𝔬_{α} ∩ K).

In the important case that the place γ is surjective, i.e. γ(k) = L, we would like to choose the places μ_{s} in such a way that (μ_{s}(K_{t(s)}) | 0 ≤ s ≤ e) is a generic splitting tower of \overline{φ}. For not necessarily surjective γ, one could try to achieve that (L · μ_{s}(K_{t(s)}) | 0 ≤ s ≤ e) is a generic splitting tower of \overline{φ}, where L · μ_{s}(K_{t(s)}) denotes the subfield of L_{s} generated by L and μ_{s}(K_{t(s)}) in L_{s}.^{21} In any case, for arbitrary choice of μ_{s} – as above – the following theorem holds.

Theorem 1. For t(s − 1) < r ≤ t(s) the subfield μ_{s}(K_{r}) · L of L_{s}, generated by μ_{s}(K_{r}) and L, is a generic splitting field of \overline{φ} of level s.

Proof. Let F_{r}: = λ_{r}(K_{r}) · L. By restricting the range, we obtain from λ_{r} a place from K_{r} to F_{r} which extends γ. Hence

ind(\overline{φ} ⊗ F_{r}) ≥ ind(φ ⊗ K_{r}) = i_{0} + … + i_{r}.

Since ind(\overline{φ} ⊗ F_{r}) is a number of SP(\overline{φ}), we get

ind(\overline{φ} ⊗ F_{r}) ≥ i_{0} + … + i_{t(s)}.

Thus there is a place from L_{s} to F_{r} over L. Since F_{r} ⊂ L_{s}, we get F_{r} ∼_{L} L_{s}. Thus F_{r} is a generic splitting field of \overline{φ} of level s.

This theorem does not imply that (K_{t(s)} · L | 0 ≤ s ≤ e) is a generic splitting tower of \overline{φ}. However, in the following we will produce more special situations for which this is the case.

Theorem 2. There exists a regular^{22} generic splitting tower (L_{s} | 0 ≤ s ≤ e) of \overline{φ} and a place λ: K_{h} → L_{e} ∪ ∞ having the following properties:

(i) λ extends γ.

(ii) For every s ∈ {0, 1, …, e} and every r with t(s − 1) < r ≤ t(s) we have L_{s} = λ(K_{r}) · L.

(iii) L_{0} = L.

Proof. We inductively construct fields L_{s} ⊃ L and places μ_{s}: K_{t(s)} → L_{s} ∪ ∞ as follows.

s = 0: We let L_{0} := L. We choose for μ_{0} any place from K_{t(0)} to L which extends γ. This is possible by §12, Scholium 2. Obviously we have μ_{0}(K_{r})·L = L for every r with 0 ≤ r ≤ t(0).

s → s + 1: By the induction hypothesis, (L_{j} | 0 ≤ j ≤ s) is the beginning of a generic splitting tower of \overline{φ}. In particular, L_{s} is a generic splitting field of \overline{φ} of level s. Furthermore there is a place μ_{s}: K_{t(s)} → L_{s} ∪ ∞. According to §12 the form φ_{t(s)} has FR with respect to μ_{s} and (μ_{s})_{∗}(φ_{t(s)}) ≅ ψ_{s}.

21Now we are not using the notation established in §13, Definition 6. 22cf. §13, Definition 5.

§14 Generic Splitting Towers of a Specialized Form 109

Now we extend μ_{s} to a place μ̃_{s} from K_{t(s)}(φ_{t(s)}) to L_{s}(ψ_{s}) (cf. §9, the proof of Theorem 3). Let u: = t(s) + 1 and let α: K_{u} → K_{t(s)}(φ_{t(s)}) ∪ ∞ be a place over K_{t(s)}, which does indeed exist since K_{u} is a generic zero field of φ_{t(s)}. Then we have a place μ̃_{s} ∘ α from K_{u} to L_{s}(ψ_{s}) and define L_{s+1} to be the composite of the fields L_{s} and (μ̃_{s} ∘ α)(K_{u}) in L_{s}(ψ_{s}),

L_{s+1}: = L · (μ̃_{s}α)(K_{u}).

Let ν_{s}: K_{u} → L_{s+1} ∪ ∞ be the place obtained from μ̃_{s} ∘ α by restricting the range. We have

ind(\overline{φ} ⊗ L_{s+1}) ≥ ind(φ ⊗ K_{u}) = i_{0} + … + i_{u} > i_{0} + … + i_{t(s)},

and thus ind(\overline{φ} ⊗ L_{s+1}) ≥ i_{0} + … + i_{t(s+1)}. This implies

ind(ψ_{s} ⊗ L_{s+1}) ≥ i_{u} + … + i_{t(s+1)} > 0.

Thus ψ_{s} ⊗ L_{s+1} is isotropic. Since L_{s+1} ⊂ L_{s}(ψ_{s}), we conclude that L_{s+1} is a generic zero field of ψ_{s}. By §12, Scholium 2(3) we now have that ν_{s} can be extended to a place from K_{t(s+1)} to L_{s+1}. We choose μ_{s+1}: K_{t(s+1)} → L_{s+1} ∪ ∞ to be such a place. Since L_{s+1} = ν_{s}(K_{u}) · L, we have L_{s+1} = μ_{s+1}(K_{r}) · L for every r with u ≤ r ≤ t(s).

Eventually our construction yields a generic splitting tower

(L_{s} | 0 ≤ s ≤ e)

of \overline{φ} and a place λ: = μ_{e} from K_{h} to L_{e} which satisfy properties (i) and (ii) of the theorem.

Let us check that the tower (L_{s} | 0 ≤ s ≤ e) is regular! For 0 ≤ s ≤ e − 1 our construction gives L_{s} ⊂ L_{s+1} ⊂ L_{s}(ψ_{s}). If dim ψ_{s} > 2, then L_{s}(ψ_{s}) is regular over L_{s}, and thus L_{s+1} is regular over L_{s}. If dim ψ_{s} = 2, then we must have s = e − 1 and t(s) = h − 1. Next, we modify the last step of the construction above in the sense that we replace K_{h−1}(φ_{h−1}) by the algebraic closure E of K_{h−1} in this field and also K_{e−1}(ψ_{e−1}) by the algebraic closure F of K_{e−1} in K_{e−1}(ψ_{e−1}).

Let μ: = μ_{h−1}. We have φ_{h−1} ≅ ⟨a⟩ ⊗ [{1 \atop 1} {1 \atop b}] with a ∈ 𝔬_{μ}^{∗}, b ∈ 𝔬_{μ}^{∗}, and

ψ_{e−1} ≅ ⟨\overline{a}⟩ ⊗ [{1 \atop 1} 1 ; \overline{b}] with \overline{a}: = μ(a), \overline{b}: = μ(b). We obtain E = K_{h−1}(ξ)

and F = L_{e−1}(η) with generators ξ, η which satisfy the minimal equations ξ^{2} + ξ + b = 0, η^{2} + η + \overline{b} = 0. The extension K_{h}/E is inessential. This can be verified as in the proof of §13, Theorem 3 towards the end. (There for the case dim φ = 2.)

The place μ = μ_{e−1} from K_{h−1} to L_{e−1} has exactly one extension μ̃_{e−1}: E → F ∪ ∞ with μ̃_{e−1}(ξ) = η (cf. §9, Lemma 2). If we prepend a place from K_{h} to E over E to μ̃_{e−1}, we obtain a place μ_{e}: K_{h} → F ∪∞ which extends μ_{e−1}. The field L_{e}: = μ_{e}(K_{h}) · L differs from L_{e−1} since it splits the form \overline{φ} totally. Hence we must have L_{e} = F. Our generic splitting tower (L_{s} | 0 ≤ s ≤ e) is regular.

What we have obtained by now can perhaps best be understood in the important special case where the place γ: k → L∪∞ is surjective (i.e. γ(k) = L). For the given generic splitting tower (K_{r} | 0 ≤ r ≤ h) of φ we have constructed a place λ: K_{h} → L_{e} ∪ ∞ which extends γ and for which the tower (λ(K_{r}) | 0 ≤ r ≤ h) is an almost regular generic splitting tower of \overline{φ} “with repetitions”, i.e. a generic splitting tower in which the storeys are listed possibly more than once: λ(K_{r}) = λ(K_{t(s)}) for t(s − 1) < r ≤ t(s). The place λ furnishes a connection between the tower (K_{r}| 0 ≤ r ≤ h) and the generic splitting tower (L_{s}|0 ≤ s ≤ e) of \overline{φ}, L_{s} = λ(K_{t(s)}) which reflects in an obvious way how the splitting behaviour of \overline{φ} over L is a coarsening of the splitting behaviour of φ over K.

In many situations it is more natural or, for some given problem, more useful (see e.g. [KR]) to associate to the tower (K_{r} | 0 ≤ r ≤ h) a different generic splitting tower of \overline{φ}. Such a situation (with γ trivial) was depicted in §13. Next we give a further construction which applies to an arbitrary place γ with respect to which φ has FR and with \overline{φ} = γ_{∗}(φ) nondegenerate, as above.

Specifically, let (K_{r} | 0 ≤ r ≤ h) be the projective standard tower of φ (see §13, Definition 4). We define a field tower (L_{r} | 0 ≤ r ≤ h) and a sequence of places (λ_{r}: K_{r} → L_{r} ∪ ∞ | 0 ≤ r ≤ h) with L_{r} ⊃ L and L_{r} = λ_{r}(K_{r}) · L for all r, 0 ≤ r ≤ h, as follows.^{23}

We start with L_{0}: = L, λ_{0}: = γ. If λ_{r}: K_{r} → L_{r} ∪ ∞ is already defined and r < h, then φ_{r} has FR with respect to λ_{r} by §12. Let \overline{φ}_{r}: = (λ_{r})_{∗}(φ_{r}). We set L_{r+1}: = L_{r}\{\overline{φ}_{r}\}. Let λ̃_{r}: K_{r}(φ_{r}) → L_{r}(\overline{φ}_{r}) ∪ ∞ be an extension of the place λ_{r}, obtained using the procedure in the proof of §9, Theorem 3.

Next, let \overline{φ}_{r} ≅ H̸. The place λ̃_{r} maps K_{r+1} = K_{r}\{φ_{r}\} into L_{r+1} ∪ ∞. From the construction in §9 it is clear that L_{r+1} = λ̃_{r}(K_{r+1}) · L. We define λ_{r+1}: K_{r+1} → L_{r+1} ∪ ∞ by restricting the place λ̃_{r}. We have L_{r+1} = λ_{r+1}(K_{r+1}) · L.

Finally, let \overline{φ}_{r} ≅ H (and so r = h − 1). Now we have L_{r+1} = L_{r}. From the construction in §9, we have λ̃_{r}(K_{r}(φ_{r})) · L = L_{r}. Again we define λ_{r+1}, i.e. λ_{h}, to be the restriction of λ̃_{r} to a place from K_{r+1} to L_{r+1} = L_{r}. Then, clearly, λ_{r+1}(K_{r+1}) · L = L_{r+1}.

Definition. We call the tower (L_{r} | 0 ≤ r ≤ h) the transfer of the projective standard tower by the place γ and we call λ: = λ_{h}: K_{h} → L_{h}∪∞ a transferring place.

Remark. By the remark in §9, following the proof of Theorem 3, the places λ_{r}: K_{r} → L_{r} ∪ ∞ for 0 ≤ r ≤ h − 1 in the construction above are uniquely determined by γ. If \overline{φ}_{h−1} ≅ H̸ this is also the case for λ_{h} = λ. If \overline{φ}_{h−1} ≅ H, then there are at most two possibilities for λ_{h}, since K_{h} is a quadratic extension of K_{h−1} in this situation. The fields L_{r} = λ_{r}(K_{r})·L are all uniquely determined by φ and γ.

Theorem 3. Let (L_{r} | 0 ≤ r ≤ h) be the transfer of the projective standard tower (K_{r} | 0 ≤ r ≤ h) of φ by the place γ: k → L∪∞ and let λ: K_{h} → L_{h}∪∞

23The equation L_{r} = λ_{r}(K_{r}) · L only signifies that L_{r} is generated as a field by both subfields λ_{r}(K_{r}) and L.

be a transferring place. Let \overline{φ}_{r}: = (λ_{r})_{∗}(φ_{r}) be the specialization of φ_{r} with respect to the restriction λ_{r}: K_{r} → L_{r} ∪ ∞ of λ (0 ≤ r ≤ h).

Claim: (L_{r} | r ∈ J) is a generic splitting tower of \overline{φ} and for every r ∈ J we have \overline{φ}_{r} = ker(\overline{φ} ⊗ L_{r}). For r ∉ J, L_{r+1}/L_{r} is purely transcendental. In particular, the extension L_{t(0)}/L is purely transcendental.

Proof. Let r ∈ {0, …, h} be given. From the Witt decomposition

φ ⊗ K_{r} ≅ (i_{0} + … + i_{r}) × H ⊥ φ_{r}

we obtain

\overline{φ} ⊗ L_{r} ≅ (i_{0} + … + i_{r}) × H ⊥ \overline{φ}_{r},

since (λ_{r})_{∗}(φ ⊗ K_{r}) ≅ \overline{φ} ⊗ L_{r}.

If r ∉ J, then ind(\overline{φ} ⊗ L_{r}) > i_{0} + … + i_{r}. Now \overline{φ}_{r} is isotropic and hence L_{r+1} = L_{r}\{\overline{φ}_{r}\} is purely transcendental over L_{r}.

Let a number s be given with 0 ≤ s ≤ e. Then L_{t(s)+1} = L_{t(s)}\{(\overline{φ}_{t(s)})\} and L_{t(s+1)}/L_{t(s)+1} is purely transcendental. Hence L_{t(s+1)}/L_{t(s)} is a generic zero field of \overline{φ}_{t(s)}. Furthermore, L_{t(0)}/L is purely transcendental. Thus

ind(\overline{φ} ⊗ L_{t(0)}) = ind(\overline{φ}) = i_{0} + … + i_{t(0)}.

This shows that \overline{φ}_{t(0)} is the kernel form of \overline{φ} ⊗ L_{t(0)}. Suppose now that for some s ∈ \{1, …, e\} we already showed that \overline{φ}_{t(k)} is the kernel form of \overline{φ}⊗L_{t(k)} for 0 ≤ k < s. Then (L_{t(k)} | 0 ≤ k ≤ s) is the beginning of a generic splitting tower of \overline{φ}. Hence \overline{φ} ⊗ L_{t(s)} has index i_{0} + … + i_{t(s)}, and it follows that \overline{φ}_{t(s)} is the kernel form of \overline{φ} ⊗ L_{t(s)}. Thus, this holds for all s ∈ \{0, …, e\}. Now it is clear that (L_{t(s)} | 0 ≤ s ≤ e) is a generic splitting tower of \overline{φ}.

112 Chapter II. Generic Splitting Theory

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Chapter III

Some Applications

114 Chapter III. Some Applications

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§15 Subforms which have Bad Reduction

The theory of weak specialization, developed in §3, §7 and the end of §11, played until now only an auxiliary role, which we could have done without when dealing with quadratic forms (due to §11, Theorem 2). For the first time we now come to independent applications of weak specialization.

Let λ: K → L ∪ ∞ be a place and φ a bilinear or quadratic form over K having good reduction with respect to λ. If φ contains subforms which have bad reduction, there are consequences for the form λ_{∗}(φ) which we will discuss now. We will look at bilinear forms first.

Theorem 1. Let φ be a nondegenerate bilinear form over K, having good reduction with respect to λ. Suppose that there exists a nondegenerate subform ⟨b_{1}, …, b_{m}⟩ of φ with λ(b_{i}c^{2}) = 0 or ∞ for every i ∈ {1, …, m} and every c ∈ K^{∗}. Then λ_{∗}(φ) has Witt index ≥ \{\frac{m}{2}\}. (As usual, \{\frac{m}{2}\} denotes the smallest integer ≥ \frac{m}{2}.)

Proof. Let ψ: = ⟨b_{1}, …, b_{m}⟩. We have a decomposition φ ≅ ψ ⊥ η, to which we apply the additive map λ_{W}: W (K) → W (L). We get λ_{W} (ψ) = 0.^{24} Hence {λ_{∗}(φ)} = λ_{W} (φ) = λ_{W} (η). Let ρ be the anisotropic form in the Witt class λ_{W} (η). Then λ_{∗}(φ) ∼ ρ and dim ρ ≤ dim η = dim φ − m. Therefore, ind λ_{∗}(φ) ≥ \{\frac{m}{2}\}.

Corollary. Let φ be a nondegenerate bilinear form over K, having good reduction with respect to λ. Suppose that λ_{∗}(φ) is anisotropic. Then every subform of φ is nondegenerate and has good reduction with respect to λ.

Proof. Clearly φ is now also anisotropic. Therefore every subform of φ is anisotropic and so definitely nondegenerate. By Theorem 1, φ does not contain any one-dimensional subforms ⟨b⟩ with λ(bc^{2}) = 0 or ∞ for every c ∈ K^{∗}. Hence every subform of φ has good reduction.

Remark. If char L = 2, then λ_{∗}(φ) is in general only determined by λ and φ up to stable isometry. However, if a representative of λ_{∗}(φ) is anisotropic, then λ_{∗}(φ) is uniquely determined up to genuine isometry, since λ_{∗}(φ) is up to isometry the only anisotropic form in the Witt class λ_{W} (φ) in this case.

Next we can derive the following statement, which is similar to the Substitution Principle in §3, Theorem 5.

24We use the shorter notation λ_{W} (ψ) instead of λ_{W} ({ψ}), cf. §7.

Theorem 2. [K_{4}, Prop. 3.3]. Let (f_{ij}(t)) be an n × n-matrix of polynomials f_{ij}(t) ∈ k[t] over an arbitrary field k, where t = (t_{1}, …, t_{r}) is a set of indeterminates over k. Let g_{1}(t), …, g_{m}(t) be further polynomials in k[t]. Finally, let c = (c_{1}, …, c_{r}) be an r-tuple of coordinates in a field extension L of k, such that det(f_{ij}(c)) ≠ 0, and such that c is a non-singular zero of every polynomial g_{p}(t) (1 ≤ p ≤ m), i.e. g_{p}(c) = 0, \frac{∂g_{p}}{∂t_{q}} (c) ≠ 0 for an element q ∈ {1, …, n}, which depends on p. If ⟨g_{1}(t), …, g_{m}(t)⟩ is a subform of the bilinear form (f_{ij}(t)) over k(t), then the bilinear form (f_{ij}(c)) over L has Witt index ≥ \{\frac{m}{2}\}.

Proof [K_{4}, p.291]. Going from k[t] to L[t], we may suppose without loss of generality that L = k.

For r = 1, Theorem 2 follows immediately from Theorem 1, applied to the place λ: k(t_{1}) → k ∪ ∞ over k with λ(t_{1}) = c_{1}.

Now let r > 1. We first assume that k is an infinite field. Then there exists an r-tuple (a_{1}, …, a_{r}) ∈ k^{r} with

∑_{q=1}^{r} a_{q} \frac{∂g_{p}}{∂t_{q}} (c) ≠ 0

for 1 ≤ p ≤ m. By applying an obvious coordinate transformation, we obtain that \frac{∂g_{p}}{∂t_{1}} (c) ≠ 0 for every p ∈ \{1, …, m\}.

Let c′ be the (r−1)-tuple (c_{2}, …, c_{r}). We choose a place α: k(t) → k(t_{1})∪ ∞ over k(t_{1}) with α(t_{i}) = c_{i} for 2 ≤ i ≤ r. Applying the additive map α_{W}: W (k(t)) → W (k), just as in the proof of Theorem 1, shows that the space (f_{ij}(t, c′)) over k(t_{1}) is Witt equivalent to a space

⟨g_{1}(t_{1}, c′), …, g_{m}(t_{1}, c′), h_{1}(t_{1}), …, h_{s}(t_{1})⟩

with polynomials h_{p}(t_{1}), 1 ≤ p ≤ s, and m + s ≤ n. {Possibly s = 0.} Let λ: k(t_{1}) → k ∪ ∞ be the place over k with λ(t_{1}) = c_{1}. We have λ_{W} (⟨g_{p}(t_{1}, c′)⟩) = 0 for 1 ≤ p ≤ m. Then, after applying λ_{W}, we see that the form (f_{ij}(c)) over k is Witt equivalent to a form of dimension ≤ s. Since s ≤ n − m, (f_{ij}(c)) certainly has Witt index ≥ \{\frac{m}{2}\}.

Finally, let us consider the case where the field k is finite. Let u be an indeterminate over k. By what we proved above, the form (f_{ij}(c)) over k(u) has Witt index ≥ \{\frac{m}{2}\}. Upon specializing with an arbitrary place β: k(u) → k ∪ ∞ over k, we see that (f_{ij}(c)) over k has Witt index ≥ \{\frac{m}{2}\}.

Remark. If char k ≠ 2, then the part of the proof dealing with r > 1, k infinite, can be shortened by first applying the Substitution Principle §3, Theorem 5, followed by Theorem 1. If char k = 2 however, when applying §3, Theorem 5, we need that the form (f_{ij}(t_{1}, c′)) over k(t_{1}) is anisotropic, which is not necessarily the case.

If we now want to obtain theorems for quadratic forms, similar to Theorems 1 and 2, at best with fair instead of good reduction, then we cannot use weak specialization as easily as above. For instance, let φ, ψ be strictly regular forms over a field K with ψ < φ, and let λ: K → L ∪ ∞ be a place. We then have an orthogonal decomposition φ ≅ ψ ⊥ η. Suppose now that φ has FR (= fair reduction) with respect to λ and that ψ is weakly obedient (see §11, Definition 4), but far from having FR. We would like to apply the “operator” λ_{W} to the relation φ ≅ ψ ⊥ η in order to learn something about λ_{∗}(φ) = λ_{W} (φ). However, this is only possible when η is also weakly obedient with respect to λ, something we cannot assume just like that. To circumnavigate this cliff, we will not simply apply the theory about λ_{W} of §7 and §11, but rather orientate ourselves towards the argumentation which led to this theory. For quadratic forms we will obtain theorems similar, but not exactly analogous, to Theorem 1 and 2. Broadly speaking we will make more assumptions, but will also be able to draw stronger conclusions from these.

Thus, let λ: K → L ∪ ∞ be a place, 𝔬 = 𝔬_{λ} the associated valuation ring and 𝔪 = 𝔪_{λ} its maximal ideal. As before (§7) we choose a system of representatives S of Q(K)/Q(𝔬) in K^{∗} with 1 ∈ S.

In two of the following three theorems, we will make statements for quadratic and bilinear forms in parallel. Throughout we will give the proofs for the more complicated quadratic case. In the bilinear case, the proofs go in the same manner.

Theorem 3. Let E = (E, q) be a quadratic (resp. bilinear) space over K which has FR (resp. GR) with respect to λ. Further, let F be a linear subspace of E with λ(q(x)) = 0 or ∞ (resp. λ(B(x, x)) = 0 or ∞) for every x ∈ F. Then λ_{∗}(E) has Witt index ≥ dim F.

Proof in the quadratic case. We may assume that L coincides with the residue class field k = 𝔬/𝔪 of 𝔬 and that λ is the canonical place from K to k associated to 𝔬. Let m: = dim F. In E we choose a reduced nondegenerate quadratic 𝔬-module M with E = KM. Let N be the 𝔬-module M ∩ F. Then M = N ⊕ N′ where N′ is a further 𝔬-module.

The image \overline{N} of N in M/𝔪M is a k-vector space of dimension m. By our assumption about F, we have q(x) ∈ 𝔪 for every x ∈ N, and so \overline{q}(\overline{N}) = 0 where \overline{q} is the quadratic form on \overline{M}: = M/𝔪M induced by q. The quadratic space \overline{M} = (\overline{M}, \overline{q}) = λ_{∗}(E) is nondegenerate by assumption. Since QL(\overline{M}) is anisotropic, we have \overline{N} ∩ QL(\overline{M}) = \{0\}. We choose a decomposition \overline{M} = QL(\overline{M}) ⊕ G = QL(\overline{M}) ⊥ G with \overline{N} ⊂ G. The quadratic space G is strictly regular and contains the totally isotropic subspace \overline{N} of dimension m. Therefore ind G ≥ m (cf. §5, Lemma 2), and so ind \overline{M} ≥ m.

Theorem 4. Let φ be a quadratic form over K, having FR with respect to λ: K → L ∪ ∞ and containing a subform ψ of dimension m, which is weakly obedient with respect to λ and such that λ_{W} (ψ) ∼ 0. Then λ_{∗}(φ) has Witt index ≥ \{\frac{m}{2}\}.

Proof. By going from K to the henselisation K^{h} with respect to 𝔬 and accordingly from λ to the place λ^{h}: K^{h} → L ∪ ∞, we may assume without loss of generality that 𝔬 itself is henselian. We choose a weakly λ-modular decomposition of ψ,

(∗)  ψ ≅ ⊥_{s∈S} sψ_{s}

where the ψ_{s} are forms which all have FR with respect to λ.

For every s ∈ S with s ≠ 1 we have a Witt decomposition ψ_{s} ≅ ψ_{s}^{∘} ⊥ r_{s}×H with r_{s} ≥ 0, ψ_{s}^{∘} anisotropic. Thus

ψ ≅ (ψ_{1} ⊥ (∑_{s≠1} r_{s}) × H) ⊥ ⊥_{s≠1} sψ_{s}^{∘}.

Upon replacing (∗) by this weakly λ-modular decomposition of ψ, we may assume at the outset that the forms ψ_{s} with s ≠ 1 in (∗) are all anisotropic. {Of course ψ_{s} = 0 for almost all s.}

The assumption λ_{W} (ψ) = 0 signifies that λ_{∗}(ψ_{1}) ∼ 0. In particular, λ_{∗}(ψ_{1}) does not have a quasi-linear part ≠ 0. Therefore ψ_{1} is strictly regular and actually has GR with respect to λ. We have a decomposition φ ≅ ψ_{1} ⊥ η. By §11, Theorem 4, η also has FR with respect to λ and λ_{∗}(φ) ≅ λ_{∗}(ψ_{1}) ⊥ λ_{∗}(η). Since λ_{∗}(ψ_{1}) is hyperbolic, we have

ind λ_{∗}(φ) = \frac{1}{2} dim ψ_{1} + ind λ_{∗}(η).

Furthermore, ρ: = ⊥_{s≠1} sψ_{s} is a subform of η. If we can verify the statement

of the theorem for η and ρ instead of for φ and ψ, then the statement will follow for φ and ψ as well. Hence we assume from now on without loss of generality that ψ_{1} = 0.

Let E = (E, q) be a space, isometric to φ and F = (F, q|F) a subspace of E, isometric to ψ. By an old argument from §7, we have that for no vector x ∈ F the value q(x) can lie in 𝔬^{∗} because, 𝔬 being henselian, the form ψ_{s} can only represent elements c^{2}ε with c ∈ K, ε ∈ 𝔬^{∗} for every s ≠ 1, cf. §6, Theorem 5. By Theorem 3, we then have ind φ ≥ dim ψ. So, obviously ind φ ≥ \{\frac{1}{2} dim ψ\}.

Theorem 5. Let φ be a quadratic form over K which has FR with respect to λ. Suppose that the specialization λ_{∗}(φ) is anisotropic. Then every subform ψ of φ, which is weakly obedient with respect to λ, also has FR. {Note that therefore λ_{∗}(ψ) is a subform of λ_{∗}(φ) by §11, Theorem 2.}

Proof. Again we may assume that λ is the canonical place associated to 𝔬 and that 𝔬 is henselian. Just as in the previous proof, we see that ψ has a weakly λ-modular decomposition

ψ ≅ ⊥_{s∈S} sψ_{s},

where the forms λ_{∗}(ψ_{s}) for s ≠ 1 are all anisotropic. Now ⊥_{s≠1} sψ_{s} is a subform

of ψ to which we can apply Theorem 3. Since λ_{∗}(φ) is anisotropic, this form has to be the zero form. Thus ψ ≅ ψ_{1}. Hence ψ has FR with respect to λ.

Theorem 6 (cf. [K_{4}, Prop.3.5]). Let (f_{ij}(t)) be a symmetric n × n-matrix of polynomials f_{ij}(t) ∈ k[t] over an arbitrary field k, where t = (t_{1}, …, t_{r}) is a set of indeterminates. Let g_{1}(t), …, g_{m}(t) be further polynomials in k[t]. Finally, let c = (c_{1}, …, c_{r}) be an r-tuple of coordinates in a field extension L of k such that all g_{p}(c) = 0 (1 ≤ p ≤ m) and such that the m × r-matrix (\frac{∂g_{p}}{∂t_{q}} (c)) has rank m.

(i) If the quadratic form [f_{ij}(c)] over L is nondegenerate, and if the quasilinear form [g_{1}(t), …, g_{m}(t)] over k(t) is a subform of the quadratic form [f_{ij}(t)], then the form [f_{ij}(c)] over L has Witt index ≥ m.

(ii) If the bilinear form (f_{ij}(c)) over L is nondegenerate and if the bilinear form ⟨g_{1}(t), …, g_{m}(t)⟩ over k(t) is a subform of (f_{ij}(t)), then the form (f_{ij}(c)) over L has Witt index ≥ m.

Proof of (i) [K_{4}, p.292 ff.]. By going from k[t] to L[t], we may assume without loss of generality that k = L. Next we replace the indeterminates t_{i} by t_{i} −c_{i}, thus assuming without loss of generality that c = 0. Finally we subject the variables t_{i} to a linear transformation with coefficients in k and obtain

(∗)  \frac{∂g_{p}}{∂t_{q}} (0) = δ_{pq} (1 ≤ p ≤ m, 1 ≤ q ≤ r).

Now we introduce the field K = k(t, s) with indeterminate s over k(t) as well as the elements u_{i} = \frac{t_{i}}{s} (1 ≤ i ≤ r) and the field k(u) = k(u_{1}, …, u_{r}). We have K = k(u, s). Let λ: K → k(u) ∪ ∞ be the place over k(u) with λ(s) = 0. Its valuation is discrete of rank 1. It associates to every polynomial f(u, s) ∈ k[u, s] the order ord_{s}f(u, s) ∈ ℕ_{0}. This is the biggest number k ∈ ℕ_{0} such that s^{k} divides the polynomial f(u, s) in k[u, s] or in k(u)[s]. Let (F, q) be a quadratic space, isometric to [g_{1}(t), …, g_{m}(t)]. We will show:

(∗∗) λ(q(x)) = 0 or ∞ for every x ∈ F.

By Theorem 3, this will then imply that the form [f_{ij}(0)] over k(u) has Witt index ≥ m. By the Substitution Principle §11, Theorem 3, the form [f_{ij}(0)] will also have Witt index ≥ m over k. {Note that [f_{ij}(0)] obviously has the same Witt index over k and k(u).}

We come to the proof of statement (∗∗). Let x ∈ F and x ≠ 0. Then q(x) = ZN^{−1}, where Z, N are polynomials in k[u, s] of the following form:

N = h(u, s)^{2}, Z = ∑_{i=1}^{m} a_{i}(u, s)^{2}g_{i}(u_{1}s, …, u_{r}s).

Here h(u, s) and the a_{i}(u, s) are polynomials in k[u, s] with h(u, s) ≠ 0 and not all a_{i}(u, s) = 0.

Now we interpret all these polynomials as polynomials in s with coefficients in k[u] and consider the lowest terms of N and S. We have an l ∈ ℕ_{0} with

a_{i}(u, s) = b_{i}(u)s^{l} + higher terms

and at least one coefficient b_{i}(u) ≠ 0 (1 ≤ i ≤ m). By (∗), g_{i}(u_{1}s, …, u_{r}s) has u_{i}s as lowest term. Hence Z has c(u)s^{2l+1} as lowest term, with

c(u): = ∑_{i=1}^{m} b_{i}(u)^{2}u_{i},

provided that c(u) ≠ 0. This is the case however, for one can easily see that the quadratic form [u_{1}, …, u_{m}] over k(u) is anisotropic. (Note that m ≤ r.) The anisotropy is also clear by the example just after §7, Theorem 5.

Therefore ord_{s}Z = 2l + 1. On the other hand, ord_{s}N = 2 ord_{s}h is even. We conclude that (∗∗) holds.

§16 Some Forms of Height 1

In this section k can be any field. However, we are mostly interested in fields of characteristic two. For any nondegenerate quadratic form φ of height h ≥ 1 (i.e. φ not split) over k, the last but one kernel form φ_{h−1} has height 1. Thus, in order to gain an understanding of quadratic forms, it is of prime importance to have an overview of all anisotropic forms of height 1.

If char k ≠ 2, it is well-known (cf. the end of §4) and it will be shown in §20 and §23 that an anisotropic form φ over k has height 1 iff there exists an element a ∈ k^{∗} and a Pfister form τ over k with φ ≅ aτ, dim τ ≥ 2, in case dim φ is even, φ ≅ aτ′, dim τ ≥ 4, in case dim φ is odd. Here τ′ denotes the pure part of τ, cf. the end of §4. Moreover, in both cases τ is uniquely determined by φ up to isomorphism.

Pfister forms can also be defined in case k has characteristic 2 (see below). We will see that diverse forms of height 1 can be obtained from quadratic Pfister forms in this case as well. Nonetheless, the question remains open what other anisotropic quadratic forms of height 1 there are. This is currently perhaps the most important unsolved problem about quadratic forms in characteristic 2.

Definition 1.

(a) Let d ∈ ℕ_{0}. A bilinear d-fold Pfister form over k is a form

τ = ⟨1, a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{d}⟩

with coefficients a_{i} ∈ k^{∗}. {We let τ = ⟨1⟩, in case d = 0.}

(b) Let d ∈ ℕ. A quadratic d-fold Pfister form over k is a form

ρ ⊗ [{1 \atop 1} {1 \atop b}] ,

where ρ is a bilinear Pfister form of degree d − 1 and b ∈ k, 1 − 4b ≠ 0.^{25}

If τ is a bilinear d-fold Pfister form with d ≥ 1, then we have an orthogonal decomposition τ ≅ ⟨1⟩ ⊥ τ′. It is well-known [Ba_{2}, p.257] that also in characteristic 2, the form τ′ is uniquely determined by τ up to isomorphism.^{26}

25The form [1] is thus not considered to be a quadratic Pfister form, not even if

char k ≠ 2. 26If τ is anisotropic, this follows easily from §20, Theorem 8 below.

We call τ′ the pure part of the Pfister form τ. It seems that for a quadratic Pfister form ρ in characteristic 2, it does not make any sense to speak of a pure part of ρ (however, see §26, Definition 2).

Next we will discuss some elementary properties of Pfister forms.

Definition 2. If φ is a quadratic or bilinear form over k, then D(φ) denotes the set of all elements a ∈ k^{∗} which are values of the form φ, a = φ(c_{1}, …, c_{n}) for an n-tuple (c_{1}, …, c_{n}) ∈ k^{n}.^{27} We denote by N(φ) the set of all a ∈ k^{∗} with aφ ≅ φ. We call the a ∈ D(φ) the elements represented by φ and the a ∈ N(φ) the similarity norms of φ.

Note. D(φ) is the set of all a ∈ k^{∗} with ⟨a⟩ < φ in the bilinear case and [a] < φ in the quadratic case. Obviously, N(φ) is a subgroup of k^{∗} which contains all squares of elements of k^{∗}. Furthermore, D(φ) is a union of cosets of N(φ) in k^{∗}.

We call N(φ) the norm group of φ. Analogously, we define for a quadratic or bilinear module E over k the norm group N(E) and the set D(E). For instance, if E = (E, q) is a quadratic module, then D(E) is the set of all a ∈ k^{∗} for which there exists an x ∈ E with q(x) = a, and N(E) is the set of all a ∈ k^{∗} such that there exists a similarity σ: E ⟶^{∼} E with norm a, i.e. a linear automorphisms σ of E with q(σ(x)) = aq(x).

Theorem 1. If τ is a bilinear or quadratic Pfister form over k, then D(τ) = N(τ).

For this – in characteristic ≠ 2 – well-known theorem, due to Pfister, Witt has given a particularly short proof which belongs to the standard repertoire of all modern books on the algebraic theory of quadratic forms (e.g. [S, p. 69 ff.], [P, p. 27]). The proof can be extended to characteristic 2 without effort. We recall it here for the convenience of the reader.

Proof of Theorem 1. We use induction on the degree d of τ. In the bilinear case the start of the induction, d = 0, is trivial. If d = 1 in the quadratic case, then τ is the hyperbolic plane [{0 \atop 1} {1 \atop 0}] or the norm form of a quadratic separable field extension of k. In both cases the statement is clear.

Next we deduce d from d − 1. We write τ = ρ ⊥ aρ, where ρ is a (bilinear or quadratic) Pfister form of degree d − 1. Since 1 ∈ D(τ), we clearly have N(τ) ⊂ D(τ). It is also clear that N(ρ) ⊂ N(τ). Now, let c ∈ D(τ) be given. We write c = ρ(x) + aρ(y) with vectors x, y ∈ k^{m}, m = 2^{d−1}. If ρ(x) = 0 or ρ(y) = 0, then c is a similarity norm of τ since a is one and since D(ρ) = N(ρ) ⊂ N(τ) by our induction hypothesis. Suppose now that ρ(x) and ρ(y) are both nonzero. Then we have:

27If φ is bilinear, and c ∈ k^{n}, we define φ(c) := φ(c, c).

τ = ρ ⊥ aρ ≅ ρ(x)ρ ⊥ aρ(y)ρ ≅ ⟨ρ(x), aρ(y)⟩ ⊗ ρ
≅ ⟨ρ(x) + aρ(y), aρ(x)ρ(y)(ρ(x) + aρ(y))⟩ ⊗ ρ
= (ρ(x) + aρ(y))⟨1, aρ(x)ρ(y)⟩ ⊗ ρ
≅ (ρ(x) + aρ(y))⟨1, a⟩ ⊗ ρ = (ρ(x) + aρ(y))τ.

Here we several times used the fact that ρ(x) and ρ(y) are similarity norms of ρ.

Theorem 2.

(i) Let τ be an isotropic bilinear Pfister form over k. Then there exists a bilinear Pfister form σ over k with τ ≅ ⟨1, −1⟩ ⊗ σ. In particular, τ is metabolic.

(ii) Every isotropic quadratic Pfister form over k is hyperbolic.

Proof. (i) : We use induction on the degree d of τ. For d = 1 the statement is clear. So suppose d > 1. We choose a factorization τ ≅ ⟨1, a⟩ ⊗ ρ where ρ is a Pfister form of degree d − 1. If ρ is isotropic then, by our induction hypothesis, ρ ≅ ⟨1, −1⟩⊗γ for some Pfister form γ. Then τ ≅ ⟨1, −1⟩⊗σ with σ: = ⟨1, a⟩⊗γ. Now suppose that ρ is anisotropic. We have vectors x, y ∈ k^{m}, m = 2^{d−1}, with ρ(x) + aρ(y) = 0, but x and y not both zero. Since ρ is anisotropic, we must have ρ(x) ≠ 0 and ρ(y) ≠ 0. Theorem 1 then implies −a = ρ(x)ρ(y)^{−1} ∈ N(ρ). Therefore, τ ≅ ⟨1, a⟩ ⊗ ρ ≅ ⟨1, −1⟩ ⊗ ρ.

(ii) : The argumentation is completely analogous.

Corollary. If τ is an anisotropic quadratic Pfister form over k, then τ has height 1.

Proof. τ ⊗ k(τ) is an isotropic Pfister form over k(τ). Thus, τ ⊗ k(τ) ∼ 0 by Theorem 2.

If char k ≠ 2, τ is a quadratic Pfister form over k and φ is a subform of τ with dim φ = dim τ − 1, then we have: If φ is isotropic, then φ splits. Indeed, we have τ ≅ φ ⊥ [a] for some a ∈ k^{∗}. If φ is isotropic, then τ is isotropic, thus τ ∼ 0 by Theorem 2 and hence φ ∼ [−a]. As above, this implies: if φ is anisotropic, then φ has height 1. In characteristic 2, we cannot come to the same conclusion since φ is not an orthogonal summand of τ. As a replacement we have the following theorem:

Theorem 3. Let char k = 2. Let ρ be a bilinear Pfister form of degree d ≥ 1, let β = [{1 \atop 1} {1 \atop b}] be a binary quadratic form and let c ∈ D(β). Define φ: = ρ′ ⊗ β ⊥ [c], then:

(i) If φ is isotropic, then φ ≅ (2^{d} − 1) × H ⊥ [c].

(ii) If φ is anisotropic, then φ has height 1.

(iii) k(φ) is specialization equivalent with k(ρ ⊗ β) over k.

Proof. Let φ be isotropic. Since φ is a subform of ρ ⊗ β, the form ρ ⊗ β is isotropic. Hence, by Theorem 2,

ρ ⊗ β = (⟨1⟩ ⊥ ρ′) ⊗ β ∼ 0.

It follows that ρ′ ⊗ β ∼ β. {Note that β = −β.} Thus φ ∼ β ⊥ [c] ∼ [c], since β ⊥ [c] is isotropic and dim β = 2. This proves (i).

Next, suppose that φ is anisotropic. Then φ ⊗ k(φ) is isotropic and so φ ⊗ k(φ) splits by what we already proved. Therefore h(φ) = 1.

Finally, let K ⊃ k be a field extension with (ρ ⊗ β)_{K} isotropic.^{28} Then (ρ ⊗ β)_{K} ∼ 0. Hence, (ρ′ ⊗ β)_{K} ∼ β_{K} and φ_{K} ∼ (β ⊥ [c])_{K} ∼ [c]. It is now clear that ρ ⊗ β and φ become isotropic over the same field extensions of k. Thus k(ρ ⊗ β) is a generic zero field of φ.

The class of examples of forms of height 1, given by this theorem, can be expanded considerably, as we will show now. But first some generalities. We need the following theorem from the elementary theory of quadratic forms.

Theorem 4. Let E = (E, q) be a strictly regular space over k and let F_{1} = (F_{1}, q|F_{1}), F_{2} = (F_{2}, q|F_{2}) be quadratic submodules of E. Further, let α: F_{1} ⟶^{∼} F_{2} be an isometry. Then there exists an automorphism σ of E, i.e. an isometry σ: E ⟶^{∼} E, which extends α.

For a proof, we refer to [Bo_{1}, §4, Th.1]. If F_{1} (thus also F_{2}) is strictly regular, then Theorem 4 can also be formulated as a cancellation theorem and is then a special case of §6, Theorem 2. In characteristic ≠ 2, this is just the Witt Cancellation Theorem [W, Satz 4]. We dub Theorem 4 the “Witt Extension Theorem” (although Theorem 4 was not formulated in this generality by Witt).

Let E = (E, q) again be a strictly regular space. Let F be a quadratic submodule of E. Then

F^{⊥}: = {x ∈ E | B_{q}(x, E) = 0}

is also a subspace of E. Since B_{q} is nondegenerate, we have (F^{⊥})^{⊥} = F. We call F^{⊥} the polar of F in E and write F^{⊥} = Pol_{E}(F). If F_{1} is a submodule of E (w.r.t. q), isometric to F, then Pol_{E}(F_{1}) is also isometric to Pol_{E}(F) by Theorem 4. This fact supplies us with very convenient terminology in the language of forms (instead of spaces) as follows.

Definition 3. Let φ be a quadratic form over k and τ a strictly regular form over k. Suppose that φ < τ. We define a (possibly degenerate) form ψ over k as follows: Let (E, q) = E be a space, isometric to τ and (F, q|F) = F a submodule of E, isometric to φ. Then we define ψ to be given by the submodule Pol_{E}(F) of E. ψ is up to isometry uniquely determined by φ and τ. We write ψ = Pol_{τ} (φ) and call ψ “the” polar of φ in τ.

28From now on, we sometimes write ψ_{K} instead of ψ ⊗ K for a form ψ over K, as announced in §9.

We return to the geometric language. We suppose now that k has characteristic 2. On a strictly regular space (E, q) over k we can then use symplectic geometry, since B_{q}(x, x) = 0 for every x ∈ E.

Lemma 1. Let F be a quadratic submodule of a strictly regular space E and let P = QL(F) be the quasi-linear part of F. Then we also have QL(Pol_{E}(F)) = P. More precisely: let G be a complement of P in F, i.e. F = P ⊕ G = P ⊥ G. Then there exists an orthogonal decomposition E = M ⊥ G ⊥ R with P ⊂ M, dim M = 2 dim P, F = P ⊥ G. For every such decomposition, we have Pol_{E}(F) = P ⊥ R.

Proof. The space G is strictly regular. Let T = Pol_{E}(G), i.e. E = T ⊥ G. T is also strictly regular and P ⊂ T. Let x_{1}, …, x_{r} be a basis of the vector space P. Since P is quasi-linear, we have B(x_{i}, x_{j}) = 0 for all i, j ∈ {1, …, r}. By elementary symplectic geometry, T contains vectors y_{1}, …, y_{r} with B(x_{i}, y_{j}) = δ_{ij} for all i, j ∈ {1, …, r}. Let M be the vector space spanned by x_{1}, …, x_{r}, y_{1}, …, y_{r}. The bilinear form B|M ×M is clearly nondegenerate. Hence the quadratic submodule M of T is nondegenerate, i.e. is a quadratic space, and we have T = M ⊥ R with a further quadratic space R. We have E = M ⊥ R ⊥ G.

Now let E = M ⊥ R ⊥ G be a given decomposition, with P ⊂ M, dim M = 2 dim P, F = P ⊥ G. Then

Pol_{E}(F) = Pol_{E}(P) ∩ Pol_{E}(G) = Pol_{E}(P) ∩ (M ⊥ R) = Pol_{M}(P) ⊥ R.

By elementary symplectic geometry, we get Pol_{M}(P) = P, and so Pol_{E}(F) = P ⊥ R.

In the language of forms, the lemma allows us to say the following.

Scholium. Let τ be a strictly regular quadratic form and φ an arbitrary quadratic form with φ < τ. Then there is an orthogonal decomposition τ = μ ⊥ γ ⊥ ρ with

QL(φ) < μ, dim μ = 2 dim QL(φ), φ ≅ QL(φ) ⊥ γ, Pol_{τ} (φ) ≅ QL(φ) ⊥ ρ.

If QL(φ) ≅ [a_{1}, …, a_{r}], then μ ≅ [{a_{1} \atop 1} {1 \atop b_{1}} ] ⊥ … ⊥ [{a_{r} \atop 1} {1 \atop b_{r}} ] with further elements b_{1}, …, b_{r} ∈ k. We have Pol_{τ} (Pol_{τ} (φ)) = φ. If φ is nondegenerate, then Pol_{τ} (φ) is nondegenerate, since φ and Pol_{τ} (φ) have the same quasilinear part.

Later on, we will be particularly interested in the case where φ itself is quasi-linear. Then γ = 0. We thus have a decomposition τ = μ ⊥ ρ with φ < μ, dim μ = 2 dim φ, and so Pol_{τ} (φ) = φ ⊥ ρ.

Remark. The scholium remains valid in characteristic ≠ 2. Then QL(φ) is simply the radical of φ, hence a_{1} = … = a_{r} = 0. Then we can also establish that b_{1} = … = b_{r} = 0. If φ is nondegenerate (i.e. QL(φ) = 0), then Pol_{τ} (φ) is nondegenerate and τ = φ ⊥ Pol_{τ} (φ). In characteristic ≠ 2 the notion of polar is not as interesting as in characteristic 2.

Theorem 5. Again let char k = 2. Let τ be a quadratic Pfister form over k of degree d ≥ 2 and let χ be an anisotropic quasi-linear form over k with dim χ < 2^{d−1}. Suppose that χ < aτ for some a ∈ k^{∗}. Then, for the polar φ = Pol_{aτ} (χ), we have the following:

(i) φ is nondegenerate, QL(φ) = χ.

(ii) If τ is isotropic, then φ splits.

(iii) If φ is anisotropic, then φ has height 1.

(iv) The fields k(φ) and k(τ) are specialization equivalent over k.

(v) If dim χ < \frac{1}{4} dim τ, then for every form γ with φ ≅ χ ⊥ γ, k(γ) is also specialization equivalent to k(φ) over k.

Proof. 1) We assume without loss of generality that a = 1, thus χ < τ. We already know that (i) holds. We choose a decomposition φ = χ ⊥ γ. Then Pol_{τ} (φ) = χ and so, in particular, QL(φ) = χ. Upon applying the scholium to φ, we see that ρ = 0 there. We thus have a decomposition τ = μ ⊥ γ with χ < μ and 2 dim χ = dim μ. Since 2 dim χ < dim τ, we certainly have γ ≠ 0.

2) Now let τ be isotropic. By Theorem 2, we have τ ∼ 0. It follows that γ ∼ μ (note that μ = −μ) and then φ ∼ χ ⊥ μ. We have

χ ≅ [a_{1}, …, a_{r}], μ ≅ [{a_{1} \atop 1} {1 \atop b_{1}} ] ⊥ … ⊥ [{a_{r} \atop 1} {1 \atop b_{r}} ]

with elements a_{i}, b_{i} ∈ k. Now,

[a_{i}] ⊥ [{a_{i} \atop 1} {1 \atop b_{i}} ] ≅ [a_{i}] ⊥ [{0 \atop 1} {1 \atop b_{i}} ] ≅ [a_{i}] ⊥ H.

It follows that χ ⊥ μ ≅ χ ⊥ r × H, also φ ∼ χ. This proves (ii).

3) If φ is anisotropic, then φ ⊗ k(φ) is isotropic. Nevertheless, χ ⊗ k(φ) remains anisotropic, since k(φ) is separable over k (§10, Theorem 1). If we apply what we already proved to χ ⊗ k(φ) and τ ⊗ k(φ), we see that φ ⊗ k(φ) splits. Hence h(φ) = 1.

4) If φ is isotropic, then the field extensions k(φ) and k(τ) of k are both purely transcendental. Now suppose that φ is anisotropic. φ⊗k(τ) is isotropic (even split). Of course, τ ⊗ k(φ) is also isotropic. Hence k(φ) ∼_{k} k(τ) in this case also.

5) Finally, suppose that dim χ < \frac{1}{4} dim τ = 2^{d−2}. Now dim μ < 2^{d−1}, thus dim γ > 2^{d−1}. We have τ ⊗ k(τ) ∼ 0, hence γ ⊗ k(τ) ∼ μ ⊗ k(τ). Since dim μ < dim γ, it follows that γ ⊗ k(τ) is isotropic. Clearly τ ⊗ k(γ) is also isotropic. We conclude that k(γ) ∼_{k} k(τ).

We want to understand the connection between the forms φ and aτ in Theorem 5 better. For this purpose we give another general definition for quadratic forms over a field k of characteristic 2.

Definition 4. Let φ be an arbitrary form and τ a strictly regular form over k. We say that τ is a strictly regular hull of φ, and write φ <_{s} τ, when the following conditions hold:

(a) φ < τ.

(b) Let (E, q) = E be a space, isometric to τ and let (F, q|F) = F be a quadratic submodule of E, isometric to φ. Then there does not exist a strictly regular subspace (E′, q|E′) = E′ of E with F _{{⊂ \atop ≠}} E′ _{{⊂ \atop ≠}} E.

By the Witt Extension Theorem (Theorem 4), the relation φ <_{s} τ only depends on the isometry classes of φ and τ, and is independent of the choice of the spaces E and F.

Lemma 2. Let φ be an arbitrary quadratic form and τ a strictly regular quadratic form with φ < τ. Then τ is a strictly regular hull of φ if and only if Pol_{τ} (φ) is quasi-linear.

Proof. Let F ⊂ E be quadratic modules associated to φ and τ, as indicated in Definition 4. If τ is not a strictly regular hull of φ, then E contains a strictly regular subspace E′ with F ⊂ E′ and E′ ≠ E. We have an orthogonal decomposition E = E′ ⊥ E′′. Clearly, E′′ ⊂ Pol_{E}(F). The space E′′ is ≠ 0 and strictly regular. Hence Pol_{E}(F) is not quasi-linear.

Now suppose that φ <_{s} τ. We choose decompositions F = P ⊥ G, E = M ⊥ G ⊥ R which have the properties exhibited in Lemma 1. We have Pol_{E}(F) = P ⊥ R. Since M ⊥ G is strictly regular and contains the space F, we must have M ⊥ G = E, i.e. R = 0. It follows that Pol_{E}(F) = P is quasi-linear.

Remark. The definition of strictly regular hull is also meaningful in characteristic ≠ 2 and Lemma 2 still holds (a quasi-linear form then looks like [0, …, 0]), but again this term is far less interesting now. If φ and τ are both nondegenerate, then φ <_{s} τ holds only if φ ≅ τ.

Definition 5. Let φ be a quadratic form over k. We call φ a close Pfister neighbour^{29} if φ is nondegenerate and not quasi-linear and if there exist a Pfister form τ and an a ∈ k^{∗} with φ <_{s} aτ. More precisely, we then call φ a close neighbour of τ.

If φ is a close neighbour of τ and if φ is anisotropic, then Theorem 5 tells us that φ has height 1 and that k(τ) is a generic zero field of φ.

The relation “strictly regular hull” is compatible with taking tensor products, as we will show now.

Lemma 3. Let φ be an arbitrary quadratic form and τ a strictly regular quadratic form over k with φ < τ. Let σ be a bilinear nondegenerate form over k. Then σ ⊗ φ < σ ⊗ τ (this is clear) and Pol_{σ⊗τ} (σ ⊗ φ) ≅ σ ⊗ Pol_{τ} (φ).

29In §22 we will introduce the more general notion of Pfister neighbour.

Proof. Let (E, q) = E be a quadratic space, isometric to τ and (F, q|F) = F a submodule, isometric to φ. Furthermore, let S be a bilinear space, isometric to σ. By Lemma 1, we have orthogonal decompositions F = P ⊥ G, E = Q ⊥ G ⊥ R with P = QL(F) ⊂ Q and dim Q = 2 dim P. From those we obtain analogous decompositions

S ⊗ F = (S ⊗ P) ⊥ (S ⊗ G), S ⊗ E = (S ⊗ Q) ⊥ (S ⊗ G) ⊥ (S ⊗ R).

Clearly, S⊗P is the quasi-linear part of S⊗F and dim(S⊗Q) = 2 dim(S⊗P). By Lemma 1, we get

Pol_{E}(F) = P ⊥ R  and  Pol_{S⊗E}(S ⊗ F) = (S ⊗ P) ⊥ (S ⊗ R).

Therefore, S ⊗ Pol_{E}(F) = Pol_{S⊗E}(S ⊗ F).

Theorem 6. If φ <_{s} τ and if σ is a nondegenerate bilinear form, then σ ⊗ φ <_{s} σ ⊗ τ.

Proof. By Lemma 2, Pol_{τ} (φ) is quasi-linear. Using Lemma 3, it follows that Pol_{σ⊗τ} (σ ⊗ φ) = σ ⊗ Pol_{τ} (φ) is quasi-linear. Again by Lemma 2, we have σ ⊗ φ <_{s} σ ⊗ τ.

Corollary. If φ is a close neighbour of a quadratic Pfister form τ, and if σ is a bilinear Pfister form, then σ ⊗ φ is a close neighbour of the Pfister form σ ⊗ τ.

Thus it is shown that for a large class of forms of height 1, if φ is any such form and σ is any bilinear Pfister form, then the tensor product σ ⊗ φ again has height 1, insofar it is anisotropic. Such a phenomenon does not happen in characteristic ≠ 2. If φ is an anisotropic form of height 1 and not of the form aτ for some Pfister form τ, then (as mentioned above) we know in characteristic ≠ 2 that φ has the form aτ′, where τ′ is the pure part of some Pfister form. If σ is a further Pfister form ≠ ⟨1⟩, then σ ⊗ φ does not have height 1, since neither dim(σ ⊗ φ) nor 1 + dim(σ ⊗ φ) is a 2-power in this case.

The following question suggests itself: Let φ be a close neighbour of some Pfister form τ. Is τ uniquely determined by φ (up to isometry)?

We will show that this is indeed so. We will also derive other theorems about Pfister forms. However, for this purpose, we need more tools from the general theory of quadratic forms, namely the so-called Subform Theorem and a “Norm Theorem”. Both theorems are well-known in characteristic ≠ 2 and still do hold in characteristic 2. We will derive them next, see §17 and §19.

§17 The Subform Theorem

The so-called Subform Theorem of Cassels and Pfister (e.g. [S, Th.3.7]) is one of the most important tools in the algebraic theory of quadratic forms over fields of characteristic not two. In 1969/70, Pfister’s Egyptian student Mahmud Amer proved the Subform Theorem in characteristic two in his dissertation in a manner which does not leave anything to be desired. Unaware of this fact, R. Baeza proved a slightly more special version of the Subform Theorem in characteristic two in 1974 [Ba_{3}]. Unfortunately Amer’s proof was never published. In Pfister’s book [P] one can find a sketch of Amer’s proof which leaves a gap however. Since the Subform Theorem will play an important role in what follows, and will furnish us with a description of all strictly regular forms of height one in particular, we will now present Amer’s proof in its completeness. Apart from this, the paper [Ba_{3}] on this topic is also worth reading in our opinion.

Theorem 1 (“Cassels’s Lemma”). Let k be a field of arbitrary characteristic, x an indeterminate over k and p ∈ k[x] a polynomial. Let φ be a quadratic form in n variables over k which represents the polynomial p over the field k(x). Then φ already represents the polynomial p over the ring k[x], i.e. there exist polynomials f_{1}, …, f_{n} ∈ k[x] with φ(f_{1}, …, f_{n}) = p.

Proof. 1) We may assume that p ≠ 0. We may further assume that φ is nondegenerate upon replacing φ by φ̂ (cf. §6, Definition 9). If φ is isotropic, then [{0 \atop 1} {1 \atop 0}] is a subform of φ (cf. §6, Lemma 4). Clearly [{0 \atop 1} {1 \atop 0}] already represents every polynomial over k[x]. Thus we assume from now on that φ is anisotropic.

2) We choose a representation

(1)  p = φ (\frac{f_{1}}{f_{0}} , \frac{f_{2}}{f_{0}} , …, \frac{f_{n}}{f_{0}} )

with f_{i} ∈ k[x] (0 ≤ i ≤ n), f_{0} ≠ 0 and deg f_{0} minimal. (“deg” denotes the degree of a polynomial.) We assume that deg f_{0} > 0. This should lead to a contradiction.

Consider the form ψ: = ⟨−p⟩ ⊥ φ in n + 1 variables over k(x). For every polynomial f_{i} with 1 ≤ i ≤ n, let

f_{i} = f_{0}g_{i} + r_{i},

the division with remainder of f_{i} by f_{0}, thus with g_{i}, r_{i} ∈ k[x], deg r_{i} < deg f_{0}.

It can happen that r_{i} = 0, i.e. that deg r_{i} = −∞. However, for at least one i ∈ {1, …, n} we have r_{i} ≠ 0. We introduce the vectors f: = (f_{0}, f_{1}, …, f_{n}) and g: = (g_{0}, g_{1}, …, g_{n}) with g_{0}: = 1. By (1) we have ψ(f) = 0. By the minimality of deg f_{0} we certainly have ψ(g) ≠ 0 however. In particular, the vectors f and g are linearly independent over k(x).

In what follows we will denote the “inner product” B_{ψ}(f, g) simply by fg and we will use such a notation (as in Amer) also for forms different from ψ. Let

u: = ψ(g)f − (fg)g.

We have

ψ(u) = ψ(g)^{2}ψ(f) + (fg)^{2}ψ(g) − (fg)^{2}ψ(g) = 0,

since ψ(f) = 0. From the linear independence of f and g it follows that u ≠ 0.

The form ψ is also isotropic over k(x). Hence the first coordinate u_{0} of u is likewise different from zero. We have

u_{0} = ψ(g)f_{0} − fg = \frac{1}{f_{0}} [ψ(g)f_{0}^{2} − (fg)f_{0}] = \frac{1}{f_{0}} ψ(f − f_{0}g).

Every coordinate of f −f_{0}g = (0, r_{1}, …, r_{n}) is of lower degree than f_{0}. Hence deg ψ(f − f_{0}g) ≤ 2 deg f_{0} − 2 and so deg u_{0} ≤ deg f_{0} − 2. Since ψ(u) = 0 we thus obtained a representation of p of the form (1) in which the denominator polynomial u_{0} is of lower degree than f_{0}, contradiction!

Addendum (Amer). In the situation of Theorem 1, let φ be anisotropic and p ≠ 0. Furthermore, let M be a subset of {1, …, n} and p = φ(h_{1}, …, h_{n}) a representation with h_{i} ∈ k(x) for 1 ≤ i ≤ n and even h_{i} ∈ k[x] for i ∈ M. Then there exist polynomials g_{1}, …, g_{n} ∈ k[x] with p = φ(g_{1}, …, g_{n}) and g_{i} = h_{i} for i ∈ M.

Proof. We choose a representation

(2)  p = φ (\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} )

with f_{i} ∈ k[x] (0 ≤ i ≤ n), f_{0} ≠ 0, \frac{f_{i}}{f_{0}} = h_{i} for i ∈ M and deg f_{0} minimal. By way of contradiction, suppose again that deg f_{0} > 0. Using these polynomials f_{i}, we go through the proof above again, starting from step 2. This time we have r_{i} = 0 and g_{i} = h_{i} for every i ∈ M. For i ∈ M, the vector u above has i-th coordinate

\begin{aligned}u_{i} & = ψ(g)f_{i} − (fg)g_{i} = ψ(g)f_{0}h_{i} − (fg)h_{i} \\ & = [ψ(g)f_{0} − (fg)]h_{i} = u_{0}h_{i}. \end{aligned}

Thus \frac{u_{i}}{u_{0}} = h_{i} for every i ∈ M. By the above considerations we get deg u_{0} < deg f_{0}, contradiction!

Remark. If char k ≠ 2, then the Addendum is dispensable since it can easily be obtained from Theorem 1 as follows: as usual we interpret φ as a quadratic form over the k-vector space V := k^{n} with standard basis e_{1}, …, e_{n}. Suppose without loss of generality that M = {1, …, m} for an m ≤ n and W: =

∑_{i=1}^{m} ke_{i}. The form φ_{1}: = φ|W is anisotropic and thus strictly regular. We

have an orthogonal decomposition φ ≅ φ_{1} ⊥ φ_{2}. Now φ_{2} represents f_{1}: = p − φ_{1}(h_{1}, …, h_{m}) over k(x) and thus over k[x] by Theorem 1. This yields the representation of p by φ, as given in the Addendum.

Theorem 2 (Subform Theorem). Let ψ be an anisotropic quadratic form over k and let φ be a nondegenerate quadratic form over k in m variables. Let x_{1}, …, x_{m} be indeterminates over k. Suppose that the form ψ represents the element φ(x_{1}, …, x_{m}) over the field k(x_{1}, …, x_{m}). Then φ is a subform of ψ.

The proof involves several lemmata. We assume all the time that ψ is an n-dimensional anisotropic quadratic form over k and that k has characteristic two. For char k ≠ 2 one can find the proof of the Subform Theorem in many books, in particular in [P], [S].

Lemma 1. Let a, b, c ∈ k, b ≠ 0 and let x be an indeterminate over k. Suppose that ψ represents the element ax^{2}+bx+c over k(x). Then [{a \atop b} {b \atop c}] < ψ.

Proof. By Theorem 1, there exists a vector f ∈ k[x]^{n} with ψ(f) = ax^{2}+bx+c. Since ψ is anisotropic, every coordinate of f has to be a polynomial of degree ≤ 1. Hence f = xu+v, for certain vectors u, v ∈ k^{n}. Again using the notation uv: = B_{ψ}(u, v), we then have the equation

ψ(u)x^{2} + (uv)x + ψ(v) = ax^{2} + bx + c.

Therefore, ψ(u) = a, ψ(v) = c and uv = b. Hence the value matrix of ψ|ku + kv is [{a \atop b} {b \atop c}].

Lemma 2. Let E = (E, q) be an anisotropic quadratic module over k and let u, v be vectors in E with q(u) = q(v) ≠ 0. Then there exists an isometry

σ: E ⟶^{∼} E with σ(u) = v and σ(v) = u.

Proof. Suppose without loss of generality that u ≠ v. Then u+v ≠ 0, so that q(u + v) = 2q(u) + uv = uv ≠ 0. {As before, we write uv instead of B_{q}(u, v).} The quadratic submodule ku + kv of E is therefore strictly regular and it follows that

E = (ku + kv) ⊥ F

with F: = (ku + kv)^{⊥} (cf. §5, Lemma 1). The linear map σ: E → E with σ(u) = v, σ(v) = u, σ(w) = w for all w ∈ F is the one we are looking for.

Remark. The map σ is nothing else than the reflection in the hyperplane (u + v)^{⊥}, perpendicular to the vector u + v ≠ 0 (cf. [Bo_{1}, §6, No. 4]),

σ(z) = z − \frac{(u + v)z}{q(u + v)} (u + v), for all z ∈ E.

Lemma 3. Let ψ = φ ⊥ χ be an orthogonal decomposition with φ strictly regular. Let γ be a further quadratic form over k with φ ⊥ γ < ψ. Then γ < χ.

This lemma is a special case of a much more general theorem, which deserves attention in its own right. We will prove this theorem at the end of this section.

Lemma 4. Let ψ = φ ⊥ χ with dim φ = 2 and φ strictly regular. Let x_{1}, x_{2} be indeterminates over k and d an element ≠ 0 of k. Suppose that ψ represents the polynomial φ(x_{1}, x_{2}) + d over k(x_{1}, x_{2}). Then d ∈ D(χ).^{30}

Proof. Let φ = [{a \atop b} {b \atop c}] and dim ψ = n. We have b ≠ 0. Therefore, evaluating ψ at the first two standard vectors e_{1} = (1, 0, …, 0), e_{2} = (0, 1, 0, …, 0) of the vector space k^{n} yields the values ψ(e_{1}) = a, ψ(e_{2}) = c, e_{1}e_{2} = b. Let us now work in the vector space V = K^{n} over the field K: = k(x_{2}) which we also equip with the form ψ. (More precisely, we are dealing with the form ψ ⊗ K here, but we simply write ψ.) By our assumption, ψ represents the element

φ(x_{1}, x_{2}) + d = ax_{1}^{2} + (bx_{2})x_{1} + (cx_{2}^{2} + d)

over K(x_{1}). Lemma 1 tells us now that there exist vectors e′_{1}, h′ in V with ψ(e′_{1}) = a, ψ(h′) = cx_{2}^{2} + d, e′_{1}h′ = bx_{2}. By Lemma 2 there is an isometry σ: V ⟶^{∼} V with respect to ψ with σ(e′_{1}) = e_{1}. Letting h: = σ(h′) ∈ V, we then have

ψ(e_{1}) = a, ψ(h) = cx_{2}^{2} + d, e_{1}h = bx_{2}.

If h_{1}, …, h_{n} are the coordinates of h, i.e. h = ∑_{i=1}^{n} h_{i}e_{i}, then the last equation

gives e_{1}h = h_{2}b = x_{2}b, and so h_{2} = x_{2}. {Bear in mind that e_{i}e_{j} = 0 for i ≤ 2, j > 2, and uu = 0 for every u ∈ V.} By the Addendum there exists a vector g = (g_{1}, …, g_{n}) ∈ k[x_{2}]^{n} with g_{2} = h_{2} = x_{2} and ψ(g) = cx_{2}^{2} + d. Since ψ is anisotropic, all polynomials g_{i} have degree ≤ 1. Thus, g = x_{2}u + v with vectors u = (u_{1}, …, u_{n}), v = (v_{1}, …, v_{n}) in k^{n} and u_{2} = 1, v_{2} = 0. We have

cx_{2}^{2} + d = ψ(u)x_{2}^{2} + (uv)x_{2} + ψ(v),

so that ψ(u) = c, uv = 0 and ψ(v) = d. Furthermore, e_{1}u = u_{2}b = b, e_{1}v = v_{2}b = 0. Therefore ψ|ke_{1} + ku ≅ [{a \atop b} {b \atop c}] and v ∈ (ke_{1} + ku)^{⊥}. This shows that

[{a \atop b} {b \atop c}] ⊥ [d] < ψ = [{a \atop b} {b \atop c}] ⊥ χ.

By Lemma 3 we get [d] < χ.

Lemma 5. Let ψ = φ ⊥ χ with φ strictly regular, dim φ = 2r and let x_{1}, …, x_{2r} be indeterminates over k. Further, let d be an element ≠ 0 of

30One recalls the notation in §16, Definition 2.

k such that ψ represents the element φ(x_{1}, …, x_{2r}) + d over k(x_{1}, …, x_{2r}). Then d ∈ D(χ).

Proof. After a linear coordinate transformation of the variables x_{1}, …, x_{2r} we may assume that

φ = [{a_{1} \atop b_{1}} {b_{1} \atop c_{1}} ] ⊥ … ⊥ [{a_{r} \atop b_{r}} {b_{r} \atop c_{r}} ]

with elements a_{i}, b_{i}, c_{i} ∈ k, b_{i} ≠ 0. The form ψ represents the element

∑_{i=1}^{r}(a_{i}x_{2i−1}^{2} + b_{i}x_{2i−1}x_{2i} + c_{i}x_{2i}^{2}) + d over k(x_{1}, …, x_{2r}). The statement now

follows from Lemma 4 by induction on r.

Lemma 6. Let φ = [d_{1}, …, d_{m}] be an anisotropic quasi-linear form over k. Assume that there is a form ψ which represents the element φ(x_{1}, …, x_{m}) = d_{1}x_{1}^{2}+…+d_{m}x_{m}^{2} over the field k(x_{1}, …, x_{m}) with indeterminates x_{1}, …, x_{m} over k. Then φ < ψ.

Proof. By induction on m. Let m = 1. Then ψ represents the element d_{1} over k(x_{1}). Using the Substitution Principle §11, Theorem 3, it follows that ψ represents the element d_{1} over k as well.

Suppose now that m > 1. By the Substitution Principle we immediately get that d_{m} ∈ D(ψ). We choose a vector u ∈ k^{n} with ψ(u) = d_{m}. Let d: = d_{1}x_{1}^{2}+…+d_{m−1}x_{m−1}^{2}. By Theorem 1, ψ represents the element d+d_{m}x_{m}^{2} over K[x_{m}] with K: = k(x_{1}, …, x_{m−1}). Since ψ is anisotropic, we have vectors w′, u′ ∈ K^{n} with ψ(x_{m}u′ + w′) = d_{m}x_{m}^{2} + d. This implies

ψ(u′) = d_{m}, ψ(w′) = d, u′w′ = 0.

By Lemma 2, the vector space K^{n} has an isometry σ: K^{n} ⟶^{∼} K^{n} with respect to ψ with σ(u′) = u. Letting w: = σ(w′) ∈ K^{n}, we then have

ψ(u) = d_{m}, ψ(w) = d, uw = 0.

Thus, the form ψ_{1}: = ψ|(ku)^{⊥} represents the element d over K, and so [d_{1}, …, d_{m−1}] < ψ_{1}, by our induction hypothesis. Hence there is a system of linearly independent vectors v_{1}, …, v_{m−1} in (ku)^{⊥} with ψ(v_{i}) = d_{i} (1 ≤ i ≤ m − 1) and v_{i}v_{j} = 0 for i ≠ j. The vector u cannot be a linear combination of v_{1}, …, v_{m−1}, for if there was a relation u = a_{1}v_{1}+…+a_{m−1}v_{m−1} with coefficients a_{i} ∈ k, applying ψ to this relation would give d_{m} = a_{1}^{2}d_{1} + … + a_{m−1}^{2}d_{m−1}, contradicting the anisotropy of φ. Thus W: = kv_{1} + … + kv_{m−1} + ku is an m-dimensional subspace of k^{n} and ψ|W ≅ [d_{1}, …, d_{m}].

Theorem 2 now follows in a straightforward way from Lemmas 5 and 6. We conclude this section with the generalization of Lemma 3, announced above.

Theorem 3. Let E and F be free quadratic modules over a valuation ring 𝔬 and let G be a strictly regular quadratic space over 𝔬. If G ⊥ F < G ⊥ E, then F < E.

Proof. We work in the quadratic module U: = G⊥E with associated quadratic form q. By our assumption, U contains a submodule V, which is a direct summand of U (as an 𝔬-module), so that V = (V, q|V) has an orthogonal decomposition V = G′ ⊥ F′ with G ≅ G′, F ≅ F′. Since G′ is strictly regular, we have a further orthogonal decomposition U = G′ ⊥ E′. This gives E ≅ E′ by the Cancellation Theorem §6, Theorem 2. Hence, there is an isometry σ: U ⟶^{∼} U with σ(G′) = G and σ(E′) = E. Since F′ is orthogonal to G′, F′′: = σ(F′) has to be orthogonal to G. Therefore, F′′ ⊂ E. The 𝔬-module σ(V) = G ⊥ F′′ is a direct summand of U = G ⊥ E, and so the 𝔬-module E/F′′ ≅ U/σ(V) is free. Thus F′′ is a direct summand of E, and so F < E.

Remark. The statement of Theorem 3 remains valid for a local ring 𝔬 because the Cancellation Theorem §6, Theorem 2 holds more generally over local rings.

§18 Milnor’s Exact Sequence

Let k be an arbitrary field. We want to derive an exact sequence, going back to J. Milnor [M_{2}], which establishes a close relationship between the Witt ring W (k(t)) of the rational function field k(t) in one variable t and the Witt rings of field extensions of k, generated by one element. We need this sequence in order to prove the announced Norm Theorem in §19. It should be emphasized that we will deal with nondegenerate bilinear forms now, although we are mostly interested in quadratic forms where applications are concerned. Milnor assumes in [M_{2}, §5] that char k ≠ 2, but this restriction is not necessary here.

Let P be the set of monic irreducible polynomials p(t) ∈ k[t]. For every p ∈ P, let k_{p} denote the residue class field k[t]/(p) and λ_{p}: k(t) → k_{p} ∪ ∞ the canonical place with valuation ring 𝔬_{p}: = k[t]_{(p(t))}. {Note: except for the ring 𝔬_{∞}: = k[t^{−1}]_{(t^{−1})}, these rings are exactly all nontrivial valuation rings of k(t) which contain the field k. They are all discrete.}

For every polynomial p ∈ P and every nondegenerate (symmetric) bilinear form φ over k(t), let ∂_{p}(φ) denote the weak specialization of the form ⟨p⟩φ = ⟨p⟩ ⊗ φ,

∂_{p}(φ): = (λ_{p})_{W}(⟨p⟩φ) ∈ W (k_{p}),

(cf. §3). The Witt class ∂_{p}(φ) can be described in an elementary way for onedimensional forms as follows: Let f(t) ∈ k[t] be a polynomial ≠ 0. Consider the decomposition f(t) = p(t)^{m}g(t), with m ∈ ℕ_{0} and g(t) not divisible by p(t). If m is even, then ∂_{p}(⟨f⟩) = 0. If m is odd, then ∂_{p}(⟨f⟩) = ⟨\overline{g}⟩, where \overline{g} denotes the residue class of g in k_{p}. \{Actually, we should write [⟨\overline{g}⟩] instead of ⟨\overline{g}⟩, but we avoid such intricacies, now and in the future. From now on we will usually not make a notational distinction between a form and its associated Witt class.}

Thus we obtain an additive map

∂_{p}: W (k(t)) ⟶ W (k_{p}).

Furthermore we have a natural ring homomorphism

i: W (k) ⟶ W (k(t)), φ ⟼ φ ⊗ k(t).

W (k(t)) is a W (k)-algebra by means of this ring homomorphism. Analogously, we interpret every ring W (k_{p}), p ∈ P, as a W (k)-algebra. In what follows, only the structure of W (k(t)) and the W (k_{p}) as modules over W (k) will be important however. The maps ∂_{p} are W (k)-linear. We combine them in a W (k)-linear map

∂ = (∂_{p}|p ∈ P): W (k(t)) ⟶ ⨁_{p∈P} W (k_{p}).

Theorem (Milnor [M_{2}]). The sequence of W (k)-modules

0 ⟶ W (k) ⟶^{i} W (k(t)) ⟶^{∂} ⨁_{p∈P} W (k_{p}) ⟶ 0

is split exact. If λ: k(t) ⟶ k ∪ ∞ is an arbitrarily chosen place over k, then

λ_{W}: W (k(t)) ⟶ W (k)

is a retraction for i, in other words λ_{W} ∘ i = id_{W (k)}.

We will carry out the proof in several steps, broadly following the style of Milnor’s procedure [M_{2}, p.335ff]. In what follows, a “form” is always a nondegenerate symmetric bilinear form, which will be identified with its Witt class.

1) If φ is a form over k, then φ ⊗ k(t) has good reduction with respect to every place λ: k(t) → k ∪ ∞ over k and

λ_{W} (φ ⊗ k(t)) = λ_{∗}(φ ⊗ k(t)) = φ.

This proves the last statement of the theorem. In particular, i is injective. In the following, we think of W (k) as a subring of W (k(t)) via the map i.

2) We consider an ascending filtration (L_{d} | d ∈ ℕ_{0}) of the W (k)-algebra W (k(t)). For every d ∈ ℕ_{0}, let L_{d} be the subring of W (k(t)), generated by one-dimensional forms ⟨f⟩ for polynomials f ∈ k[t] with f ≠ 0, deg f ≤ d. Clearly we then have

W (k) = L_{0} ⊂ L_{1} ⊂ L_{2} ⊂ …,

and W (k(t)) is the union of all L_{d}. The forms ⟨f_{1} … f_{s}⟩ for products of finitely many polynomials f_{1}, …, f_{s} with deg f_{i} ≤ d (1 ≤ i ≤ s) generate L_{d} additively. Further, let (U_{d} | d ∈ ℕ_{0}) be the filtration of ⨁_{p∈P} W (k_{p}) with

U_{0} : = 0,  U_{d} : = ⨁_{{p∈P \atop deg p≤d}} W (k_{p})  (d > 0).

Clearly ∂(L_{d}) ⊂ U_{d} for every d ∈ ℕ_{0} and in particular, ∂(L_{0}) = 0. Therefore ∂ induces for every d ≥ 1 a W (k)-linear homomorphism

\overline{∂}_{d}: L_{d}/L_{d−1} ⟶ U_{d}/U_{d−1}.

For every d ∈ ℕ we will verify that \overline{∂}_{d} is bijective. As an immediate consequence we then obtain that for every d ∈ ℕ, ∂ induces an isomorphism from L_{d}/L_{0} to U_{d} and thus an isomorphism from W (k(t))/W (k) to ⨁_{p} W (k_{p}),

which concludes the proof.

We let P_{d} denote the set of all p ∈ P of degree d and we identify U_{d}/U_{d−1} with the direct sum of the W (k)-modules W (k_{p}), p ∈ P_{d}. The component of index p ∈ P_{d} of the map

\overline{∂}_{d}: L_{d}/L_{d−1} ⟶ ⨁_{p∈P_{d}} W (k_{p})

is the W (k)-linear map

\overline{∂}_{p}: L_{d}/L_{d−1} ⟶ W (k_{p}),

induced by ∂_{p}: W (k(t)) → W (k_{p}).

3) Proof of the surjectivity of \overline{∂}_{d} (d ≥ 1). It suffices to prove that, for given p ∈ P_{d} and η ∈ W (k_{p}), there exists a ξ ∈ L_{d} with ∂_{p}(ξ) = η, ∂_{p}(ξ) = 0 for q ∈ P_{d}, q ≠ p. Furthermore, we may assume that η is the Witt class of a one-dimensional form, i.e. η = ⟨\overline{f}⟩ for an f ∈ k[t] with deg f < d, f ≠ 0, where \overline{f} obviously denotes the image of f in k_{p} = k[t]/p(t). Letting ξ: = ⟨pf⟩ clearly accomplishes our goal.

4) For the proof of the injectivity of \overline{∂}_{d} we employ the following lemma (statement and proof are exactly as in Milnor [M_{2}, Lemma 5.5]).

Lemma. Let d ≥ 1. The additive group L_{d} is modulo L_{d−1} generated by elements ⟨pg_{1} … g_{s}⟩ with s ∈ ℕ_{0}, p ∈ P_{d} and every g_{i} a polynomial of degree < d. Moreover, if f is the polynomial of degree < d with f ≡ g_{1} … g_{s} mod p, then

⟨pf⟩ ≡ ⟨pg_{1} … g_{s}⟩ mod L_{d−1}.

Proof of the Lemma. We recall that the identity

⟨a + b⟩ = ⟨a⟩ + ⟨b⟩ − ⟨ab(a + b)⟩

holds in the Witt ring of any field, cf. §3. Let ⟨f_{1} … f_{r} g_{1} … g_{s}⟩ be a generator of L_{d} with pairwise different f_{i} ∈ P_{d} and polynomials g_{j} ≠ 0 of degree < d. If r ≥ 2, then f_{1} = f_{2} + h with h ≠ 0, deg h < d, and we obtain the identity

⟨f_{1}⟩ = ⟨f_{2}⟩ + ⟨h⟩ − ⟨f_{1}f_{2}h⟩.

Multiplying with ⟨f_{2} … f_{r} g_{1} … g_{s}⟩ and omitting quadratic factors gives

⟨f_{1} … f_{r} g_{1} … g_{s}⟩ = ⟨f_{3} … g_{s}⟩ + ⟨hf_{2} … g_{s}⟩ − ⟨f_{1}hf_{3} … g_{s}⟩.

Every term on the right-hand side contains at most r − 1 factors of degree d. By induction on r we now obtain that L_{d} modulo L_{d−1} is additively generated by forms ⟨fg_{1} … g_{s}⟩, where f is monic of degree d, and every g_{j} is a nonzero polynomial of degree < d. Furthermore, we may assume that f is irreducible.

Consider such a generator ⟨pg_{1} … g_{s}⟩ with p ∈ P_{d}, deg g_{i} < d. Assume that s ≥ 2. Then g_{1}g_{2} = pk + h with polynomials k, h of degree < d and h ≠ 0. Hence,

⟨g_{1}g_{2}⟩ = ⟨h⟩ + ⟨pk⟩ − ⟨pkhg_{1}g_{2}⟩.

Multiplying with ⟨pg_{3} … g_{s}⟩ gives

⟨pg_{1}g_{2} … g_{s}⟩ = ⟨phg_{3} … g_{s}⟩ + ⟨kg_{3} … g_{s}⟩ − ⟨khg_{1} … g_{s}⟩,

and so

⟨pg_{1} … g_{s}⟩ ≡ ⟨phg_{3} … g_{s}⟩ mod L_{d−1}.

Furthermore,

g_{1} … g_{s} ≡ hg_{3} … g_{s} mod p.

The proof of the Lemma can now be concluded by means of a simple induction on s.

5) Proof of the injectivity of \overline{∂}_{d} (d ≥ 1). Assume for the sake of contradiction that \overline{∂}_{d} is not injective. Then there exists a ξ ∈ L_{d} \ L_{d−1} with ∂_{p}(ξ) = 0 for every p ∈ P_{d}. By the Lemma, we have a congruence

ξ ≡ ∑_{i=1}^{s}⟨p_{i}⟩φ_{i} mod L_{d−1}

with finitely many p_{i} ∈ P_{d} and forms φ_{i} = ⟨f_{i1}, …, f_{i,r_{i}}⟩ for nonzero polynomials f_{ij} of degree < d. So, for every i ∈ {1, …, s},

∂_{p_{i}}(⟨p_{i}⟩φ_{i}) = ∂_{p_{i}}(ξ) = 0.

Furthermore, ∂_{q}(⟨p_{i}⟩φ_{i}) = 0 for q ∈ P_{d}, q ≠ p_{i}. Hence every summand ⟨p_{i}⟩φ_{i} of ξ modulo L_{d−1} is already in the kernel of \overline{∂}_{d}. Thus we certainly have a p ∈ P_{d} and polynomials f_{1}, …, f_{r} ∈ k[t] \ {0} of degree < d such that

ξ: = ⟨pf_{1}, …, pf_{r}⟩ ∉ L_{d−1},

but ∂_{p}(ξ) = 0, and so ∂_{q}(ξ) = 0 for every q ∈ P_{d}. This should now lead to a contradiction.

We assume that the sequence f_{1}, …, f_{r} is chosen to be of minimal length r. We have ∂_{p}(ξ) = ⟨\overline{f}_{1}, …, \overline{f}_{r}⟩. Thus, the form ⟨\overline{f}_{1}, …, \overline{f}_{r}⟩ over k_{p} is metabolic and so in particular isotropic. Hence, after permuting the \overline{f}_{i} if necessary, there certainly exists an index s ∈ {2, …, r} and polynomials u_{1}, …, u_{s} ∈ k[t] \ {0} of degree < d, such that

\overline{u}_{1}^{2}\overline{f}_{1} + … + \overline{u}_{s}^{2}\overline{f}_{s} = 0

in k_{p}. By the Lemma we have

⟨pu_{i}^{2}f_{i}⟩ ≡ ⟨ph_{i}⟩ mod L_{d−1},

where h_{i} ∈ k[t] is the polynomial of degree < d with u_{i}^{2}f_{i} ≡ h_{i} mod p. {Note: every h_{i} ≠ 0.} Certainly h_{1} + h_{2} ≠ 0, for otherwise we could write ξ modulo L_{d−1} in a shorter way, contradicting the minimality of r. Applying the relation

⟨h_{1}, h_{2}⟩ = ⟨h_{1} + h_{2}, h_{1}h_{2}(h_{1} + h_{2})⟩,

while keeping in mind that by the Lemma

⟨ph_{1}h_{2}(h_{1} + h_{2})⟩ ≡ ⟨pg_{1}⟩ mod L_{d−1}

for a polynomial g_{1} ≠ 0 of degree < d, we obtain

ξ ≡ ⟨p(h_{1} + h_{2}), pg_{1}, ph_{3}, … ph_{s}, pf_{s+1}, … pf_{r}⟩ mod L_{d−1}.

Continuing in this way, we finally obtain a congruence

ξ ≡ ⟨p(h_{1} + … + h_{s−1}), ph_{s}, pg_{1}, …, pg_{s−2}, pf_{s+1}, …, pf_{r}⟩ mod L_{d−1}.

The polynomial h_{1} + … + h_{s} is of degree < d and is divisible by p. Hence, h_{1} + … + h_{s} = 0. It follows that

ξ ≡ ⟨pg_{1}, …, pg_{s−2}, pf_{s+1}, …, pf_{r}⟩ mod L_{d−1},

which contradicts the minimality of r. This establishes the injectivity of \overline{∂}_{d} and concludes the proof of the Theorem.

140 Chapter III. Some Applications

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§19 A Norm Theorem

Let φ be a (symmetric) bilinear or quadratic form over a field k. We are interested in the norm group N(φ) of φ, which is the group of similarity norms of φ. It consists of all a ∈ k^{∗} with φ ≅ aφ and was already introduced in §16.

If φ is a quadratic form, then we have a decomposition

φ = φ̂ ⊥ r × [0]

with φ̂ nondegenerate and r ∈ ℕ_{0} (cf. §6). Clearly, N(φ) = N(φ̂). Therefore we may assume without loss of generality that φ is nondegenerate. By an analogous argument we may also assume this in the bilinear case.

Thus we always assume that φ is nondegenerate in what follows. In the quadratic case we have a Witt decomposition

φ ≅ φ_{0} ⊥ r × H

with φ_{0} anisotropic, r ∈ ℕ_{0}. Clearly, N(φ) = N(φ_{0}). Thus we may always assume that φ is anisotropic in the quadratic case. In the bilinear case and when char k = 2 we only have a stable isometry

φ ≈ φ_{0} ⊥ r × H̃,

with φ_{0} anisotropic, r ∈ ℕ_{0} and φ_{0} is up to isometry uniquely determined by φ. Now N(φ) is a subgroup of N(φ_{0}) and is possibly different from N(φ_{0}). In what follows we will almost always assume that φ is anisotropic.

Let K: = k(t) be the rational function field over k for a set of indeterminates t = (t_{1}, …, t_{r}). We want to compare the norm groups of φ and φ ⊗ K. Clearly, N(φ) is a subgroup of N(φ ⊗ k(t)). Furthermore, N(φ ⊗ k(t)) contains the group K^{∗2} of all squares f^{2} with f ∈ K, f ≠ 0. Thus N(φ ⊗ K) is obtained from the group N(φ)K^{∗2} by adjoining certain square-free polynomials f(t) ∈ k[t], i.e. polynomials f(t) ≠ 0 which are products of pairwise different irreducible polynomials. Our task will now be to determine these polynomials f(t).

We endow the monomials t_{1}^{α_{1}} … t_{r}^{α_{r}} (α_{i} ∈ ℕ_{0}) with the lexicographic ordering with respect to the sequence of variables t_{1} ≥ t_{2} ≥ … ≥ t_{r}. Thus, if (α_{1}, …, α_{r}) ∈ ℕ_{0}^{r} and (β_{1}, …, β_{r}) ∈ ℕ_{0}^{r} are two multi-indices, then the monomial t_{1}^{α_{1}} … t_{r}^{α_{r}} precedes the monomial t_{1}^{β_{1}} … t_{r}^{β_{r}} if there exists a j ∈ {1, …, r} with α_{i} = β_{i} for i < j, but α_{j} < β_{j}.

For every polynomial f ∈ k[t] with f ≠ 0, let f^{∗} ∈ k^{∗} denote the leading coefficient of f, i.e. the coefficient of the highest monomial occurring in f. Clearly we have (fg)^{∗} = f^{∗}g^{∗}. We call the polynomial f monic if f ≠ 0 and f^{∗} = 1.

The following theorem tells us, among other things, that we may restrict ourselves to monic square-free polynomials in the task above.

Theorem 1. Let φ be anisotropic. Let f(t) ∈ k[t] be a similarity norm of φ ⊗ K and let t_{1}^{m_{1}} … t_{r}^{m_{r}} be the highest monomial occurring in f. Then the exponents m_{i} are all even and f^{∗} is a similarity norm of φ.

Proof. By induction on r. Let r = 1. Write t_{1} = t. Let m: = deg f. We use the place λ: k(t) → k ∪ ∞ over k with λ(t) = ∞. {Note: the associated valuation ring is k[t^{−1}]_{(t^{−1})}.}

We choose an element a ∈ D(φ). Then af ∈ D(φ⊗k(t)). The form φ⊗k(t) has good reduction with respect to λ and λ_{∗}(φ ⊗ k(t)) = φ is anisotropic. By §15, Theorem 3, there exists a ξ ∈ k(t)^{∗} with λ(afξ^{2}) finite and ≠ 0. This shows that the degree m of f is even. It follows that

(†)  φ ⊗ k(t) ≅ ⟨t^{−m}f⟩ ⊗ (φ ⊗ k(t)).

Now λ(t^{−m}f) = f^{∗}. In particular, ⟨t^{−m}f⟩ has good reduction with respect to λ. Applying λ_{∗} to (†) gives

φ ≅ ⟨f^{∗}⟩ ⊗ φ.

This proves the theorem for r = 1. Assume that r > 1. We write t = (t_{1}, t′) with t′ = (t_{2}, …, t_{r}) and then have K = K′(t_{1}) with K′: = k(t_{2}, …, t_{r}). Applying the result for r = 1 to the anisotropic form φ ⊗ K′, we get that m_{1} is even and that the highest coefficient h ∈ k[t′] of f, as a polynomial in t_{1}, is a similarity norm of φ ⊗ K′. By the induction hypothesis, m_{2}, …, m_{r} are also even and h^{∗} = f^{∗} is a similarity norm of φ.

If p(t) ∈ k[t] is an irreducible polynomial, then k(p) denotes the quotient field of k[t]/(p), i.e. the function field of the affine hyperplane p(t) = 0.

Theorem 2 (Norm Theorem, [K_{4}] for bilinear φ, [Ba_{4}] for quadratic φ). Assume that the form φ is anisotropic. In the quadratic case, assume moreover that φ is strictly regular. Further, let p(t) ∈ k[t] = k[t_{1}, …, t_{r}] be a monic irreducible polynomial. Then the following are equivalent.

(i) p(t) ∈ N(φ ⊗ k(t)). (ii) p(t) divides a square-free polynomial f(t) ∈ k[t] with f(t) ∈ N(φ⊗k(t)). (iii) φ ⊗ k(p) ∼ 0.

Under the same assumptions as in the theorem, we immediately obtain the following Corollary. A polynomial f(t) ∈ k[t], f(t) ≠ 0 is a similarity norm of φ⊗k(t) if and only if f^{∗} ∈ N(φ) and φ ⊗ k(p) ∼ 0 for every monic irreducible polynomial p ∈ k[t] such that the exponent of p in f is odd.

The rest of this section will be devoted to the proof of Theorem 2. In the main part of the proof (showing that (iii) ⇒ (i)) we will use induction on r. A priori we note that if the theorem holds for a given fixed r ∈ ℕ, then the corollary also holds for this r.

The implication (i) ⇒ (ii) is trivial. In order to prove (ii) ⇒ (iii), we use the place λ_{p}: k(t) → k(p) ∪ ∞ over k with λ_{p}(p) = 0 (cf. § 18; there we write k_{p} instead of k(p)). By assumption we have

φ ⊗ k(t) ≅ ⟨f⟩ ⊗ (φ ⊗ k(t)).

To this isometry we apply the map (λ_{p})_{W}, i.e. weak specialization with respect to λ_{p}, as introduced in §3 in the bilinear case and in §7 in the quadratic case. We obtain φ ⊗ k(p) ∼ 0, as desired.

Now we come to the prove of the implication (iii) ⇒ (i), which will keep us busy for a longer time. Suppose that r = 1 and let φ be an anisotropic bilinear form. As before we write t: = t_{1}. We use the exact sequence from §18,

0 ⟶ W (k) ⟶^{i} W (k(t)) ⟶^{∂} ⨁_{q∈P} W (k(q)) ⟶ 0,

with retraction

(λ_{∞})_{∗}: W (k(t)) ⟶ W (k)

with respect to the place λ_{∞}: k(t) → k ∪∞ over k, which maps t to ∞. Let^{31}

z: = ⟨1, −p⟩ ⊗ φ_{k(t)} ∈ W (k(t)).

Clearly, ∂_{q}(z) = 0 for every monic irreducible polynomial q ≠ p. Furthermore, ∂_{p}(z) is the Witt class of −φ ⊗ k(p) which is also zero by assumption.

Assume that the polynomial p has degree m. By a well-known theorem of Springer (e.g. [S, Th.5.3]) which also holds for bilinear forms in characteristic two by §10, Theorem 1^{32}, φ ⊗ E is anisotropic for every finite extension E/k of odd degree. Since φ ⊗ k(p) ∼ 0 by assumption, m has to be even. Hence (λ_{∞})_{W}(z) = 0 and it follows from §18 that z = 0. Since φ is anisotropic, this gives

φ_{k(t)} ≅ ⟨p⟩ ⊗ φ_{k(t)} ,

as required.

Since we never proved Springer’s Theorem for char k ≠ 2, we will show again that m has to be even, using a different argument.

We have seen that ∂(z) = 0. Hence, by §18, there exists an anisotropic form ψ over k with z ∼ ψ_{k(t)}. Assume that m is odd. Then (λ_{∞})_{W}(z) = (λ_{∞})_{∗}(φ_{k(t)}) = φ. On the other hand, (λ_{∞})_{W}(z) = (λ_{∞})_{∗}(ψ_{k(t)}) = ψ. Therefore, ψ = φ. We obtain

31Now we will often write φ_{k(t)} instead of φ ⊗ k(t) etc. 32For φ one considers the quadratic form φ ⊗ [1].

⟨1, −p⟩ ⊗ φ_{k(t)} ∼ φ_{k(t)},

and so ⟨−p⟩ ⊗ φ_{k(t)} ∼ 0. This is absurd since φ_{k(t)} is anisotropic. Thus m has to be even. This proves the theorem for r = 1 in the bilinear case.

Let us now consider the case where r = 1 and φ is an anisotropic strictly regular quadratic form over k. By assumption we have φ ⊗ k(p) ∼ 0. From this we can deduce that p is a similarity norm of φ ⊗ k(t). In case k has characteristic ≠ 2, we already know this since φ can then be interpreted as a bilinear form.

So, suppose that char k = 2. We want to “lift” φ to a form over a field of characteristic zero in the easiest way possible, cf. [Ba_{4}]. For this purpose we choose a henselian discrete valuation ring A with maximal ideal 𝔪 = 2A ≠ 0, generated by 2, and residue class field A/𝔪 = k. That this is possible is well-known, cf. [E], [EP]. The quotient field K: = Quot A has characteristic zero.

We choose elements a_{1}, …, a_{n} in A such that

p(t): = t^{n} + \overline{a}_{1}t^{n−1} + … + \overline{a}_{n},

where \overline{a}_{i} of course denotes the image of a_{i} in A/𝔪 = k. The polynomial

p̃(t): = t^{n} + a_{1}t^{n−1} + … + a_{n}

is irreducible in A[t] and thus also irreducible in K[t]. Let

A(p̃): = A[t]/(p̃(t)).

This is a henselian discrete valuation ring with maximal ideal 2A(p̃) and residue class field

A(p̃)/2A(p̃) = k[t]/(p(t)) = k(p).

The valuation ring A(p̃) has K(p̃) as its quotient field.

Let us use the geometric language now. Suppose that the form φ corresponds to the quadratic space (E, q) over k. Here we may choose E = M/𝔪M for some free A-module M of rank m. We equip M with a strictly regular quadratic form q̃: M → A in such a way that (E, q) is the reduction of (M, q̃) modulo 𝔪, which is possible. Let Ẽ: = K ⊗_{A} M. Our form q̃ extends to a strictly regular quadratic form q̃_{K}: Ẽ → K. This gives us a strictly regular quadratic space Ẽ: = (Ẽ, q̃_{K}) over K which has good reduction with respect to the canonical place λ: K → k ∪ ∞ associated to A, and whose specialization λ_{∗}(Ẽ) is equal to the given space E = (E, q) over k. By assumption we have k(p) ⊗ E ∼ 0.

We may interpret the canonical place μ: K̃(p) → k(p) ∪ ∞ associated to the valuation ring A(p̃) as an extension of the place λ. By §8, Theorem 8, the space K̃(p) ⊗K̃ Ẽ again has good reduction with respect to μ and

μ_{∗}(K(p) ⊗ Ẽ) = λ_{∗}(k(p) ⊗ Ẽ) = k(p) ⊗ E ∼ 0.

From the quadratic space M = (M, q̃) over A we obtain a quadratic space A(p̃) ⊗_{A} M over A(p̃). The statement μ_{∗}(k(p) ⊗ Ẽ) ∼ 0 tells us that the reduction of A(p̃) ⊗_{A} M to the maximal ideal of A(p̃) is hyperbolic. Since A(p̃) is henselian, A(p̃) ⊗_{A} M itself is hyperbolic and then

K(p̃) ⊗_{K} Ẽ = K(p̃) ⊗_{A(p̃)} (A(p̃) ⊗_{A} M) ∼ 0.

Since char K ≠ 2, it follows from what we proved above (the bilinear case, r = 1) that p̃(t) is a similarity norm of K(t) ⊗_{K} Ẽ, i.e.

(∗)  K(t) ⊗_{K} Ẽ ≅ ⟨p̃⟩ ⊗ (K(t) ⊗_{K} Ẽ).

We want to specialize this isometry with respect to a suitable place from K(t) to k(t). Let S ⊂ A[t] be the multiplicatively closed set of all polynomials f ∈ A[t] for which at least one coefficient is not in the maximal ideal 𝔪 = 2A (“unimodular” polynomials). We construct the localization A(t): = S^{−1}A[t] of A[t] at this set. One can easily verify that A(t) is a discrete valuation ring with maximal ideal 2A(t), residue class field A(t)/2A(t) = k(t) and quotient field Quot A(t) = K(t). Let ρ: K(t) → k(t) ∪ ∞ be the canonical place associated to the valuation ring A(t). It extends the canonical place λ: K → k ∪∞ associated to the valuation ring A. Clearly the quadratic space K(t) ⊗_{K} Ẽ has good reduction with respect to ρ and

ρ_{∗}(K(t) ⊗_{K} Ẽ) = k(t) ⊗_{k} λ_{∗}(Ẽ) = k(t) ⊗_{k} E.

Applying ρ_{W} to the isometry (∗) above gives

k(t) ⊗_{k} E ≅ ⟨p⟩ ⊗ (k(t) ⊗_{k} E).

This shows that p is a similarity norm of φ ⊗ k(t) and we have completely proved the theorem for r = 1.

To conclude, we prove the implication (iii) ⇒ (i) for r > 1, simultaneously for φ bilinear or quadratic. We may suppose that the theorem, and thus also the corollary, is valid for r −1 variables. Also, we already established that the implications (i) ⇒ (ii) ⇒ (iii) hold for r variables. Next we will prove the implication (ii) ⇒ (i) for r variables and only then the implication (iii) ⇒ (i).

Thus, let t = (t_{1}, t_{2}, …, t_{r}) = (t_{1}, t′) with t′ = (t_{2}, …, t_{r}). Let p ∈ k[t] be monic and irreducible and let f ∈ k[t] \ {0}. Assume that the exponent of p in f is odd. We have to show that p is a similarity norm of φ_{k(t)}.

Suppose first that the field k contains infinitely many elements. Let n: = deg_{t_{1}} p be the degree of p, as a polynomial over k[t′]. We distinguish the cases n = 0, n > 0.

Let n = 0. We have a decomposition

f(t) = p(t′)h(t)

in k[t]. We choose an element c in k such that p(t′) in k[t′] does not divide the polynomial h(c, t′). This is possible since |k| = ∞. {One considers the image of h(t) in (k[t′]/(p))[t_{1}].} By the Substitution Principle (§3, Theorem 5 in the bilinear case, §11, Theorem 3 in the quadratic case) we have f(c, t′) ∈ N(φ_{k(t′)}). By the implication (ii) ⇒ (i) for r − 1 variables, it follows that p = p(t′) ∈ N(φ_{k(t′)}) and thus also p ∈ N(φ_{k(t)}).

Now let n > 0. Let a(t′) be the highest coefficient of p(t), as a polynomial in t_{1}. The polynomial a(t′) ∈ k[t′] is monic and p̃: = a^{−1}p ∈ k(t′)[t_{1}] is a monic polynomial in the variable t_{1}. In the ring k(t′)[t_{1}], the exponent of p̃ in f is odd. Thus, by the established case r = 1, p̃ is a similarity norm of φ_{k(t′)}, and hence also of φ_{k(t)}. Therefore,

(‡)  ⟨p⟩ ⊗ φ_{k(t)} ≅ ⟨a⟩ ⊗ φ_{k(t)}.

Thus, it suffices to show that a is a similarity norm of φ_{k(t′)}. Let π(t′) ∈ k[t′] be a monic irreducible polynomial whose exponent in a in the ring k[t′] is odd (if such a polynomial π exists at all). We will show that π is a similarity norm of φ_{k(t′)} and know then that π is indeed a similarity norm of φ_{k(t)}.

π(t′) does not divide any of the coefficients of p(t) as a polynomial in t_{1}. Since |k| = ∞, there exists an element c in k with π(t′) ∤ p(c, t′) in the ring k[t′]. By the Substitution Principle, we obtain from (‡) that

⟨p(c, t′)⟩ ⊗ φ_{k(t′)} ≅ ⟨a(t′)⟩ ⊗ φ_{k(t′)},

and so

a(t′)p(c, t′) ∈ N(φ_{k(t′)}).

The exponent of π(t′) in a(t′)p(c, t′) is odd. From the implication (ii) ⇒ (i) for r − 1 variables it follows that π ∈ N(φ_{k(t′)}) and we are finished.

Now suppose that k is a finite field. We consider the field k(u) with u an indeterminate. Upon applying what we proved before to φ_{k(u)} and p, f, considered as polynomials over k(u), we see that p is a similarity norm of φ_{k(u)}. By the Substitution Principle we see that p is a similarity norm of φ, upon specializing u to 0. This completely proves the implication (ii) ⇒ (i) for r variables.

Finally, we come to the proof of the implication (iii) ⇒ (i) for r variables. As before, let a(t′) be the highest coefficient of p(t) as a polynomial in t_{1} and let p̃: = a^{−1}p. Then k(p) = k(t′)(p̃) and by our assumption we have φ ⊗ k(p) ∼ 0. The implication (iii) ⇒ (i) in the case r = 1 tells us that p̃ is a similarity norm of φ_{k(t)}. Hence, a(t′)p(t) ∈ N(φ_{k(t)}). By the implication (ii) ⇒ (i) for r variables, proved above, p(t) is a similarity norm of φ_{k(t)}. The proof of Theorem 2 is now complete.

§20 Strongly Multiplicative Forms

Let φ be a (symmetric) bilinear or quadratic form over a field k in n variables and let t = (t_{1}, …, t_{n}) be a tuple of n unknowns over k.

Definition 1. The form φ is called strongly multiplicative when φ(t) is a similarity norm of the form φ_{k(t)} = φ ⊗ k(t),^{33} in other words when

⟨φ(t)⟩ ⊗ φ_{k(t)} ≅ φ_{k(t)}.

Examples. (1) Clearly every hyperbolic quadratic form r × H is strongly multiplicative.

(2) If τ is a bilinear or quadratic Pfister form over k, then τ_{k(t)} is likewise such a form over k(t). Thus §16, Theorem 1 tells us that N(τ_{k(t)}) = D(τ_{k(t)}). In particular we have τ(t) ∈ N(τ_{k(t)}). Thus every Pfister form is strongly multiplicative.

(3) If φ is a strongly multiplicative bilinear form, then φ ⊗ [1] is clearly a strongly multiplicative quasi-linear quadratic form. In particular, for every bilinear Pfister form τ, the quadratic form τ ⊗ [1] is strongly multiplicative.

The main aim of this section is to determine all strongly multiplicative bilinear and quadratic forms over k. We start with two very simple theorems whose proofs only require the Substitution Principle.

As before we use D(φ) to denote the set of elements of k^{∗} which are represented by φ, i.e. the set of all a ∈ k^{∗} with ⟨a⟩ < φ in the bilinear case and [a] < φ in the quadratic case. This set consists of cosets of the norm group N(φ) in the group k^{∗}.

Theorem 1. For a bilinear or quadratic form φ over k the following statements are equivalent:

(a) φ is strongly multiplicative.

(b) For every field extension L of k we have D(φ_{L}) ⊂ N(φ_{L}), and so D(φ_{L}) = N(φ_{L}).

Proof. (a) ⇒ (b): We have φ_{k(t)} ≅ ⟨φ(t)⟩⊗φ_{k(t)}. This implies φ_{L(t)} ≅ ⟨φ(t)⟩⊗ φ_{L(t)} and so, by the Substitution Principle, φ_{L} ≅ ⟨φ(c)⟩⊗φ_{L} for every c ∈ L^{n} with φ(c) ≠ 0.

33In the bilinear case we defined φ(t) := φ(t, t). Thus φ(t) = (φ ⊗ [1])(t).

(b) ⇒ (a): Trivial.

Theorem 2. If φ is a quadratic form with associated nondegenerate quadratic form φ̂, i.e. (cf. §6, Definition 9)

φ = φ̂ ⊥ δ(φ) = φ̂ ⊥ s × [0]

for some s ∈ ℕ_{0}, then φ is strongly multiplicative if and only if φ̂ is strongly multiplicative.

Proof. φ(t_{1}, …, t_{n}) = φ̂(t_{1}, …, t_{n−s}) is a similarity norm of φ_{k(t)} if and only if this polynomial is a similarity norm of φ̂_{k(t)}. This will be the case exactly when φ̂(t_{1}, …, t_{n−s}) is a similarity norm of φ̂_{k(t_{1},...,t_{n−s})}, as can be seen by the Substitution Principle for instance.

In the study of strongly multiplicative quadratic forms we may thus restrict ourselves to nondegenerate forms φ. By an analogous argument we may also do this in the bilinear case. Thus we always assume that φ is nondegenerate in what follows. Let us commence our study with strongly multiplicative quadratic forms.

Theorem 3. Let φ be a nondegenerate strongly multiplicative quadratic form over k. Then φ is either quasi-linear or strictly regular.

Proof. We recall that the quasi-linear part QL(φ) of φ is uniquely determined by φ up to isometry. Thus

⟨φ(t)⟩ ⊗ φ_{k(t)} ≅ φ_{k(t)}

implies that

⟨φ(t)⟩ ⊗ QL(φ)_{k(t)} ≅ QL(φ)_{k(t)}.

Assume that QL(φ) ≠ 0. We choose an element b ∈ D(QL(φ)). Then bφ(t) ∈ D(QL(φ)_{k(t)}). The form QL(φ) is anisotropic. Thus bφ is a subform of QL(φ) by the Subform Theorem of §17. Since dim φ ≥ dim QL(φ) it follows that φ = QL(φ).

Theorem 4. Let φ be a strictly regular strongly multiplicative quadratic form over k which is not hyperbolic. Assume that dim φ > 1.

(i) φ is an anisotropic Pfister form. (ii) If τ is another quadratic Pfister form over k with τ < φ, then there exists

a bilinear Pfister form σ over k with φ ≅ σ ⊗ τ.

Proof. (i) Let φ_{0} be the kernel form of φ, i.e. φ ≅ φ_{0} ⊥ r × H for some r ≥ 0. We have φ_{0} ≠ 0. Now ⟨φ(t)⟩ ⊗ φ_{k(t)} ≅ φ_{k(t)} implies that

⟨φ(t)⟩ ⊗ (φ_{0})_{k(t)} ≅ (φ_{0})_{k(t)}

and then bφ(t) ∈ D(φ_{0} ⊗ k(t)) for an arbitrarily chosen b ∈ D(φ_{0}). By the Subform Theorem of §17, bφ is a subform of φ_{0}. Therefore dim φ ≤ dim φ_{0} and so r = 0. Thus φ is anisotropic.

(ii) By Theorem 1 we have N(φ) = D(φ). In particular 1 ∈ D(φ). Since φ is strictly regular, there exists a 2-dimensional form [{1 \atop 1} {1 \atop a}] < φ. This form is a Pfister form.

Now let some Pfister form τ < φ be given. We choose a bilinear Pfister form σ of maximal dimension such that σ ⊗ τ < φ. {Note that possibly σ = ⟨1⟩.} The theorem will be proved if we can show that φ ≅ σ ⊗ τ.

By way of contradiction, assume that φ ≅ σ̸ ⊗ τ. Then we have a decomposition

φ ≅ σ ⊗ τ ⊥ ψ

with ψ ≠ 0. Assume that the Pfister form ρ: = σ ⊗ τ has dimension m (a power of 2). Let t = (t_{1}, …, t_{m}) be a tuple of m unknowns over k. Since φ is strongly multiplicative, we have by Theorem 1 that

ρ(t)ρ_{k(t)} ⊥ ρ(t)ψ_{k(t)} ≅ ρ_{k(t)} ⊥ ψ_{k(t)}.

Now ρ(t)ρ_{k(t)} ≅ ρ_{k(t)}. By the Cancellation Theorem (§6, Theorem 2) it follows that

ρ(t)ψ_{k(t)} ≅ ψ_{k(t)}.

We choose a b ∈ D(ψ) and then have bρ(t) ∈ D(ψ_{k(t)}). By the Subform Theorem we get bρ < ψ and so

(⟨1, b⟩ ⊗ σ) ⊗ τ = ⟨1, b⟩ ⊗ ρ < φ.

This contradicts the maximality of dim σ. Thus σ ⊗ τ ≅ φ.

In §16 we determined that every anisotropic quadratic Pfister form τ has height 1. The same is therefore true for aτ where a ∈ k^{∗} is an arbitrary scalar. We are now in the position to prove a converse of this fact.

Theorem 5. Let φ be an anisotropic strictly regular quadratic form over k of height 1. Choose a ∈ D(φ). Then φ ≅ aτ for some Pfister form τ.

Proof. Upon replacing φ by aφ we may assume without loss of generality that a = 1. As before let n: = dim φ and let t = (t_{1}, …, t_{n}) be a tuple of n unknowns over k. We have n ≥ 2.

The polynomial φ(t) over k is irreducible. Moreover, after an appropriate linear transformation of the variables t_{1}, …, t_{n} over k, we have that φ(t) is monic.

Since the height of φ is 1, φ⊗k(φ) ∼ 0. From the Norm Theorem of §19 it follows that φ(t) is a similarity norm of φ, thus φ is strongly multiplicative. And so, by Theorem 4, φ is a Pfister form.^{34}

34More precisely we should say “φ is isometric to a Pfister form”. In the following we frequently allow inaccuracies like this one, in order to simplify the language.

We now move on to quasi-linear strongly multiplicative (quadratic) forms. Let char k = 2 and let us contemplate arbitrary anisotropic quasi-linear forms over k. We will utilize the geometric language to this end.

Thus, let E = (E, q) be an anisotropic quasi-linear quadratic space over k. As usual, we denote by k^{\frac{1}{2}} the subfield of the algebraic closure k̃, consisting of all λ ∈ k̃ with λ^{2} ∈ k. Recall that for every a ∈ k there is a unique element \sqrt{a} ∈ k^{\frac{1}{2}} such that (\sqrt{a})^{2} = a and also that the map a ↦ \sqrt{a} is an isomorphism from the field k to k^{\frac{1}{2}} .

Clearly f: E → k^{\frac{1}{2}} , f(x): = \sqrt{q(x)} is a k-linear map from E to k^{\frac{1}{2}} . This map is injective since (E, q) is anisotropic. We equip the k-vector space k^{\frac{1}{2}} with the quadratic form sq: k^{\frac{1}{2}} → k, sq(x) = x^{2}. \{sq as in “square”.\} Then k^{\frac{1}{2}} becomes an – in general infinite-dimensional – quasi-linear quadratic kmodule and f an isometry from (E, q) to a submodule U of (k^{\frac{1}{2}} , sq). The following stronger statement is now obvious.

Theorem 6. For every anisotropic quasi-linear quadratic space (E, q) over k there exists exactly one isometry f: E ⟶^{∼} U, where U is a finite-dimensional quadratic submodule of (k^{\frac{1}{2}} , sq).

Now it is easy to classify the strongly multiplicative anisotropic quasilinear forms over k.

Theorem 7. Let φ be an anisotropic quasi-linear quadratic form over k. The following statements are equivalent:

(1) φ is strongly multiplicative.

(2) D(φ) = N(φ).

(3) There exists exactly one subfield K of k^{\frac{1}{2}} with k ⊂ K, [K: k] finite, such that the quadratic form (K, sq|K) is isometric to φ.

(4) φ ≅ τ ⊗ [1] where τ is an anisotropic bilinear Pfister form.

Proof. The implication (1) ⇒ (2) is clear by Theorem 2.

(2) ⇒ (3): By Theorem 6 there exists exactly one finite-dimensional ksubspace U of k^{\frac{1}{2}} such that the form sq|U is isometric to φ. By assumption (2) we certainly have 1 ∈ D(φ). Hence U contains an element z with z^{2} = 1. We must have z = 1 and thus k ⊂ U. Furthermore, for every two elements x, y in U there exists an element z in U such that z^{2} = x^{2} · y^{2}, and so z = xy. Finally, if x ≠ 0 is any element in U, then x^{−1} = (x^{−1})^{2} · x is also in U. We conclude that U is a subfield of k^{\frac{1}{2}} .

(3) ⇒ (4): We have K = k(\sqrt{a_{1}}, …, \sqrt{a_{d}}) with finitely many elements a_{1}, …, a_{d} in k which are linearly independent over the field k^{(2)} = {x^{2}|x ∈ k}. The associated quadratic form φ is the orthogonal sum of the one-dimensional forms [a_{i_{1}}a_{i_{2}} … a_{i_{r}}] with 1 ≤ i_{1} < i_{2} < ⋯ < i_{r} ≤ d, i.e.

φ = ⟨1, a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{d}⟩ ⊗ [1].

(4) ⇒ (1): We already established above (Example 3) that for every bilinear Pfister form τ, the form τ ⊗ [1] is strongly multiplicative.

Definition 2. Following established terminology (e.g. [Lo]) we call a quadratic or bilinear form φ over a field k round whenever N(φ) = D(φ). Analogously we call a quadratic or bilinear module E over k round whenever N(E) = D(E), thus, in finite dimensions, when the (up to isometry unique) associated form φ is round.

By Theorem 2 it is clear that a form φ over k is strongly multiplicative if and only if for every field extension L of k the form φ_{L} is round. For this to hold, it is sufficient to know that φ_{k(t)} is round where, as before, t = (t_{1}, …, t_{n}) is a tuple of n = dim φ unknowns over k.

Theorem 7 tells us, among other things, that a round quasi-linear quadratic form φ over a field of characteristic 2 is already strongly multiplicative. In Theorem 7 we supposed that φ is anisotropic, to be sure, but this assumption can subsequently be dropped.

We remark that in general there are many more round forms than strongly multiplicative forms among the non quasi-linear quadratic forms. For example if k = ℝ, then for every n ∈ ℕ the form n × [1] is round. This form is nonetheless strongly multiplicative only when n is a power of 2.

Now let us assume again that char k = 2. Among other things, Theorem 7 tells us that every anisotropic quasi-linear round quadratic space over k can be considered as a finite field extension K of k with K ⊂ k^{\frac{1}{2}} in a unique way. We want to give a second, more structural, proof of this fact. This will guide us in the right direction for a classification of the anisotropic round bilinear forms over k.

Thus, let (E, q) be an anisotropic round quasi-linear quadratic module over k. Every vector x ∈ E is uniquely determined by the value q(x) ∈ k. For if q(x) = q(y), then q(x − y) = 0 and so x − y = 0. {Note: this holds for every anisotropic quasi-linear space.} Since D(E) is a group, there exists a vector e_{0} with q(e_{0}) = 1. Furthermore there exists for every x ∈ E, x ≠ 0 exactly one similarity transformation σ_{x}: E → E with norm q(x). Hence we have for every vector y ∈ E,

q(σ_{x}y) = q(x)q(y).

We define a multiplication (x, y) ↦ x · y on E by x · y: = σ_{x}y for x ≠ 0, 0 · y: = 0. As usual we write xy instead of x · y most of the time. Clearly we then have for arbitrary x, y ∈ E that

q(xy) = q(x)q(y).

Since q(yx) = q(y)q(x) = q(xy) it follows that yx = xy. If z is a further element in E we have

q((xy)z) = q(x)q(y)q(z) = q(x(yz)),

so that (xy)z = x(yz). Finally,

q(e_{0}z) = q(e_{0})q(z) = q(z),

and so e_{0}z = z. Our multiplication is thus commutative and associative and has e_{0} as identity element. By definition it is k-linear in the second argument and thus also in the first argument. Hence E is a commutative k-algebra. If x ∈ E, x ≠ 0, then q(x) ≠ 0 and furthermore q(x · x) = q(x)^{2}, so that

q(q(x)^{−1}x · x) = 1.

It follows that q(x)^{−1}x · x = e_{0}. Thus x has an inverse x^{−1} = q(x)^{−1}x. The ring E is therefore a field. As usual we denote its identity element e_{0} from now on by 1 and identify k with the subfield k ·1 of E. In this way E becomes a field extension of k and for every x ∈ E we have

x^{2} = q(x).

If k^{\frac{1}{2}} is the set of all square roots \sqrt{a} of elements a ∈ k in a given fixed algebraic closure k̃ of k, then we have exactly one field embedding E ↪ k^{\frac{1}{2}} of E in the field k^{\frac{1}{2}} over k. Thus we have derived the implication (2) ⇒ (3) in Theorem 7 for the second time. In fact we obtained an analogous result for E of infinite dimension over k, which we actually will not need in what follows.

Now let (E, B) be a round anisotropic bilinear module over the field k of characteristic 2. The quadratic form n(x): = B(x, x) on E, associated to B, is round, quasi-linear and anisotropic. Thus E carries the structure of a field extension of k with x^{2} = n(x) for all x ∈ E.

Definition 3. We call the extension field U of k in k^{\frac{1}{2}} which is isomorphic to E over k, the inseparable field extension of k associated to (E, B).

The field U ⊂ k^{\frac{1}{2}} should be considered as a first invariant of the space (E, B). What else do we need in order to describe (E, B) completely up to isometry?

We endow the k-vector space E with the linear form

t: E → k, t(x): = B(1, x).

Clearly, t(1) = n(1) = 1. Now B is completely determined by t since for every x ∈ E, x ≠ 0 there is exactly one similarity transformation σ_{x}: E ⟶^{∼} E of norm n(x) and for y ∈ E we have σ_{x}(y) = xy, all of this being justified by our analysis of the round quasi-linear space (E, n) above. Hence we have for arbitrary y, z ∈ E

(∗)  B(xy, xz) = n(x)B(y, z).

This equality remains valid when x = 0. Let y ≠ 0, x = y^{−1}, then (∗) implies that n(y)^{−1}B(y, z) = B(1, y^{−1}z) = t(y^{−1}z), and so B(y, z) = n(y)t(y^{−1}z) = t(n(y) · y^{−1}z) = t(y^{2}y^{−1}z) = t(yz). The equality

B(y, z) = t(yz)

remains valid when y = 0.

Conversely, let U be an extension field of k with k ⊂ U ⊂ k^{\frac{1}{2}} and let t: U → k be a linear form on the k-vector space U with t(1) = 1. On U we define a symmetric bilinear form

β_{t}: U × U → k, β_{t}(x, y): = t(xy).

We have β_{t}(x, x) = t(x^{2}) = x^{2}t(1) = x^{2}. The quadratic standard form x ↦ x^{2} on U is thus associated to β_{t}. In particular, the bilinear k-module (U, β_{t}) is anisotropic. For every x ∈ U, x ≠ 0 we define the k-linear transformation

σ_{x}: U ⟶ U, y ⟼ xy.

We have

β_{t}(σ_{x}y, σ_{x}z) = β_{t}(xy, xz) = t(x^{2}yz) = x^{2}t(yz) = x^{2}β_{t}(y, z).

Thus σ_{x} is a similarity transformation on U of norm x^{2}. This shows that (U, β_{t}) is round.

We summarize these considerations in the following theorem.

Theorem 8. The round anisotropic bilinear modules over k are up to isometry the pairs (U, β_{t}) with U an extension field of k in k^{\frac{1}{2}} and t a k-linear form on U with t(1) = 1. For any round anisotropic bilinear module (E, B) over k there is exactly one such pair (U, β_{t}) with (E, B) ≅ (U, β_{t}) and furthermore exactly one isometry from (E, B) to (U, β_{t}).

Theorem 9. Let φ be an anisotropic round bilinear form over a field k of characteristic 2.

(i) φ is a Pfister form. (ii) If τ is a bilinear Pfister form with τ < φ, then there exists a further

bilinear Pfister form σ with φ ≅ σ ⊗ τ.

Proof. We have ⟨1⟩ < φ. Let τ < φ be an arbitrary Pfister form. We choose a Pfister form σ over k of maximal dimension with σ ⊗ τ < φ. {Note: it is possible that σ = ⟨1⟩.} We will show that σ ⊗ τ = φ, which will prove the theorem.

We will use the geometric language. Let E = (E, B) be a bilinear space which arises from φ and let F be a subspace of E arising from σ ⊗ τ. E carries the structure of a field extension of k with x^{2} = B(x, x) for all x ∈ E. Furthermore there is a linear form t: E → k with B(x, y) = t(xy) for all x, y ∈ E. Finally, F is a subfield of E with k ⊂ E. {If σ ⊗ τ ≅ ⟨1 ⊗ a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{m}⟩, then F = k(\sqrt{a_{1}}, …, \sqrt{a_{m}}) and t(\sqrt{a_{i_{1}} … a_{i_{r}}}) = 0 for r > 0, 1 ≤ i_{1} < ⋯ < i_{r} ≤ m.}

Assume for the sake of contradiction that F ≠ E. We choose z ∈ E with z ≠ 0, B(z, F) = 0. For x, y ∈ F we then have

B(x, zy) = t(xyz) = B(xy, z) = 0.

Thus zF is a subspace of E, perpendicular to F. Finally we have for x, y ∈ F with b: = B(z, z) the equality

B(zx, zy) = bB(x, y).

All this shows that F ⊥ zF is a subspace of E, with associated bilinear form the Pfister form ⟨1, b⟩⊗σ ⊗τ. This contradicts the maximality of dim σ. Thus F = E.

By Theorems 4 and 9 it is now clear that over every field the anisotropic strongly multiplicative bilinear forms are exactly the anisotropic Pfister forms.

§21 Divisibility by Pfister Forms

In this section we will deal with the divisibility of a quadratic form by a given quadratic or bilinear Pfister form in the sense of the following general definition.

Definition 1. Let φ and τ be quadratic forms and σ a bilinear form over k. We say that φ is divisible by τ if there exists a bilinear form ψ over k with φ ≅ ψ ⊗ τ. Similarly we say that φ is divisible by σ if there exists a quadratic form χ over k with φ ≅ σ ⊗ χ. We then write τ|φ resp. σ|φ.

First we will study the divisibility by a given (anisotropic) quadratic Pfister form.

Theorem 1. Let τ be a quadratic Pfister form over a field k and φ an anisotropic strictly regular quadratic form over k. The following statements are equivalent.

(i) τ|φ. (ii) There exists a nondegenerate bilinear form ψ over k with φ ∼ ψ ⊗ τ. (iii) φ ⊗ k(τ) ∼ 0.

Proof. The implications (i) ⇒ (ii) ⇒ (iii) are trivial.

(iii) ⇒ (i): Let dim τ = n (a 2-power) and let t = (t_{1}, …, t_{n}) be a tuple of n unknowns over k. We can think of τ(t) ∈ k[t] as a monic polynomial. By the Norm Theorem (cf. §19) our assumption φ ⊗ k(τ) ∼ 0 implies that τ(t) is a similarity norm of φ_{k(t)}. We choose an element a ∈ D(φ). Then aτ(t) ∈ D(φ_{k(t)}) and thus, by the Subform Theorem (cf. §17),

φ ≅ aτ ⊥ φ_{1}

where φ_{1} is a further quadratic form over k. It now follows that φ_{1} ⊗ k(τ) ∼ 0. By induction on dim φ we obtain φ ≅ χ ⊗ τ for a bilinear form χ = ⟨a, a_{2}, …, a_{r}⟩.

The following theorem is a variation of Theorem 1 in the case where φ is also a Pfister form.

Theorem 2. Let τ and ρ be quadratic Pfister forms over a field k. Assume that the form ρ is anisotropic. The following statements are equivalent.

(i) There exists a bilinear Pfister form σ over k with ρ ≅ σ ⊗ τ. (ii) τ|ρ. (iii) There exists an element a ∈ k^{∗} with aτ < ρ. (iv) ρ ⊗ k(τ) ∼ 0.

(v) There exists a place λ: k(ρ) → k(τ) ∪ ∞ over k.

Proof. The implications (i) ⇒ (ii) and (ii) ⇒ (iii) are trivial. The implications (iii) ⇒ (v) and (iv) ⇔ (v) both follow from the fact that k(ρ) is a generic zero field of ρ and that every isotropic quadratic Pfister form is already hyperbolic. The implication (iv) ⇒ (ii) is clear by Theorem 1. To finish the proof we verify the implication (iii) ⇒ (i). Thus let aτ < ρ for some a ∈ k^{∗}. Then a ∈ D(ρ) = N(ρ). Hence τ < aρ ≅ ρ. Now §20, Theorem 4 tells us that there exists a bilinear Pfister form σ over k with ρ ≅ σ ⊗ τ.

Corollary. Let τ_{1} and τ_{2} be quadratic Pfister forms over k such that k(τ_{1}) ∼_{k} k(τ_{2}). Then τ_{1} ≅ τ_{2}.

Note that in the proof of Theorem 1 we showed that for a given a ∈ D(φ) we could choose the factor ψ in such a way that a ∈ D(ψ). Similarly one could ask whether in Theorem 2 the Pfister form σ can be chosen so as to be divisible by ⟨1, a⟩, for an appropriately given binary form ⟨1, a⟩. This leads us to the next lemma which is very useful for what follows.

Lemma 1. Let σ be a bilinear and τ a quadratic Pfister form over k. Let dim σ > 1 and a ∈ D(σ′ ⊗ τ), where – as before (§4, §16) – σ′ denotes the pure part of σ. Then there exists a bilinear Pfister form γ over k with

σ ⊗ τ ≅ γ ⊗ ⟨1, a⟩ ⊗ τ.

Proof. (a) If σ ⊗ τ is isotropic, then σ ⊗ τ is hyperbolic and the statement can easily be verified. Namely, if dim σ > 2, then we can choose γ to be the hyperbolic Pfister form of the correct dimension. If σ = ⟨1, c⟩ however, then a = cu for some u ∈ D(τ), so that ⟨1, c⟩ ⊗ τ ≅ ⟨1, a⟩ ⊗ τ, and γ = ⟨1⟩ will do.

(b) Suppose next that σ ⊗ τ is anisotropic. Let μ: = ⟨1, a⟩ ⊗ τ and K: = k(μ). By Theorem 2 it is sufficient to show that (σ ⊗ τ)_{K} ∼ 0.

Since μ_{K} ∼ 0 we have that τ_{K} ≅ −aτ_{K} and

(σ ⊗ τ)_{K} = τ_{K} ⊥ (σ′ ⊗ τ)_{K} ≅ (−aτ ⊥ σ′ ⊗ τ)_{K}.

By assumption the element a is represented by σ′ ⊗ τ. Thus the form −aτ ⊥ σ′ ⊗ τ is isotropic. Hence (σ ⊗ τ)_{K} is isotropic. Since σ ⊗ τ is a Pfister form we have (σ ⊗ τ)_{K} ∼ 0, as required.

Now suppose that ρ_{1}, ρ_{2} are two quadratic Pfister forms over k, possibly of different dimension. We are looking for a quadratic Pfister form τ, perhaps of higher dimension, which divides both ρ_{1} and ρ_{2}. In this situation a concept emerges which was coined by Elman and Lam [EL, §4] in characteristic ≠ 2:

Definition 2. (a) Let r be a natural number. We call ρ_{1} and ρ_{2} r-linked, more precisely, quadratically r-linked, if over k there exist an r-fold quadratic Pfister form τ and bilinear Pfister forms σ_{1}, σ_{2} such that

ρ_{1} ≅ σ_{1} ⊗ τ, ρ_{2} ≅ σ_{2} ⊗ τ.

(b) We define the quadratic linkage number j(ρ_{1}, ρ_{2}) as follows: if ρ_{1} and ρ_{2} are 1-linked, then j(ρ_{1}, ρ_{2}) is the largest number r such that ρ_{1} and ρ_{2} are r-linked. Otherwise we set j(ρ_{1}, ρ_{2}) = 0.

Remark. Elman and Lam only use the term “r-linked”. In our context the adjective “quadratic” is necessary since in char k = 2 we also can, and should, inquire about bilinear Pfister forms which divide both ρ_{1} and ρ_{2}, see below.

Theorem 3. (cf. [EL, Prop.4.4] for char k ≠ 2).^{35} Assume that the quadratic Pfister forms ρ_{1}, ρ_{2} are anisotropic and 1-linked. Then the form ρ_{1} ⊥ −ρ_{2} has Witt index

ind(ρ_{1} ⊥ −ρ_{2}) = 2^{r}

with r: = j(ρ_{1}, ρ_{2}) ≥ 1. Furthermore, if τ is a quadratic s-fold Pfister form with 1 ≤ s < r, dividing both ρ_{1} and ρ_{2}, then there exists an (r − s)-fold bilinear Pfister form γ and bilinear Pfister forms σ_{1}, σ_{2} such that

ρ_{1} ≅ σ_{1} ⊗ γ ⊗ τ, ρ_{2} ≅ σ_{2} ⊗ γ ⊗ τ.

Proof. We assume that we have factorizations ρ_{1} ≅ σ_{1} ⊗τ, ρ_{2} ≅ σ_{2} ⊗τ where τ is a quadratic Pfister form and σ_{1} and σ_{2} are bilinear Pfister forms. We verify the following two statements from which the theorem readily follows:

(i) ind(ρ_{1} ⊥ ρ_{2}) ≥ dim τ.

(ii) If ind(ρ_{1} ⊥ −ρ_{2}) > dim τ, then there exists an a ∈ k^{∗} and bilinear

Pfister forms γ_{1}, γ_{2} such that
ρ_{1} ≅ γ_{1} ⊗ ⟨1, a⟩ ⊗ τ, ρ_{2} ≅ γ_{2} ⊗ ⟨1, a⟩ ⊗ τ.

(i): We have ρ_{1} ≅ τ ⊥ σ′_{1} ⊗ τ, ρ_{2} ≅ τ ⊥ σ′_{2} ⊗ τ, and thus

ρ_{1} ⊥ −ρ_{2} ≅ τ ⊗ ⟨1, −1⟩ ⊥ σ′_{1} ⊗ τ ⊥ (−σ′_{2} ⊗ τ).

(ii): Now let ind(ρ_{1} ⊥ ρ_{2}) > dim τ. By our calculation this shows that the form σ′_{1} ⊗ τ ⊥ (−σ′_{2} ⊗ τ) is isotropic. Hence there exists an a ∈ k^{∗} which is represented by both of the forms σ′_{1} ⊗ τ and σ′_{2} ⊗ τ. By Lemma 1 we have factorizations

σ_{1} ⊗ τ ≅ γ_{1} ⊗ ⟨1, a⟩ ⊗ τ, σ_{2} ⊗ τ ≅ γ_{2} ⊗ ⟨1, a⟩ ⊗ τ

35Elman and Lam do not require the assumption that j(ρ_{1}, ρ_{2}) ≥ 1, cf. our Theorem 6 below.

with bilinear Pfister forms γ_{1}, γ_{2}, as required.

Let us now deal with the problems of divisibility by a given bilinear Pfister form. Unfortunately there apparently does not (yet) exist a good counterpart to Theorem 1 above. There is one for Theorem 2 however. To derive it we need a further lemma.

Lemma 2. Let ρ be a quadratic and σ a bilinear Pfister form over k. Assume that the form ρ is anisotropic. Let τ = [{1 \atop 1} {1 \atop b}] be a binary Pfister form over k with

τ ⊥ σ′ ⊗ [1] < ρ.

Then there exists a bilinear Pfister form α over k such that

ρ ≅ α ⊗ σ ⊗ τ.

Proof. Assume that the dimension of σ is 2^{d}. Then the Pfister form μ: = σ⊗τ is of dimension 2^{d+1}. Let E = (E, q) be a quadratic space corresponding to μ and let K: = k(μ). The space K ⊗ E is hyperbolic. Hence it contains a 2^{d}-dimensional vector subspace U with q(U) = 0. Let V be a subspace of K ⊗ E corresponding to the subform (τ ⊥ σ′ ⊗ [1])_{K} of μ_{K}. It has dimension 2^{d} + 1. From the obvious inequality

dim(V ∩ U) + dim E ≥ dim V + dim U

we see that dim(V ∩ U) ≥ 1. Hence the quadratic space V is isotropic. Thus ρ ⊗ K is isotropic and so ρ ⊗ K ∼ 0. Now Theorem 2 tells us that ρ ≅ α ⊗ μ for some bilinear Pfister form α.

Theorem 4. Let σ be a bilinear Pfister form and ρ an anisotropic quadratic Pfister form over k with σ ⊗ [1] < ρ. Then there exists a quadratic Pfister form τ over k such that ρ ≅ σ ⊗ τ.

Proof. Let E = (E, q) be a quadratic space over k which corresponds to the form ρ, and let V be a subspace of E which corresponds to the subform σ⊗[1] of ρ. We have an orthogonal decomposition V = ke_{0} ⊥ V′ with q(e_{0}) = 1 and V′ a quasi-linear space corresponding to the form σ′ ⊗ [1]. The bilinear form B = B_{q} vanishes on V × V. There is a linear form s on V with s(e_{0}) = 1, s(V′) = 0. Furthermore there is a vector f_{0} ∈ E with s(v) = B(v, f_{0}) for all v ∈ V. The subspace V + kf_{0} of E corresponds to the quadratic form [{1 \atop 1} {1 \atop b}] ⊥ σ′ ⊗ [1]. This form is thus a subform of ρ. The theorem now follows from Lemma 2.

Definition 3. If φ is an anisotropic bilinear form over k and dim φ > 1, then we define k(φ): = k(φ ⊗ [1]). If φ = ⟨a_{1}, …, a_{n}⟩, then k(φ) is the quotient field of the ring k[t_{1}, …, t_{n}]/(a_{1}t_{1}^{2} + ⋯ + a_{n}t_{n}^{2}). Note that the polynomial φ(t) = a_{1}t_{1}^{2} + ⋯ + a_{n}t_{n}^{2} is irreducible since φ is anisotropic.

Theorem 5. Let σ be an anisotropic bilinear Pfister form and ρ an anisotropic quadratic Pfister form over k. Assume that σ ≠ ⟨1⟩. The following statements are equivalent.

(i) There exists a quadratic Pfister form τ over k with ρ ≅ σ ⊗ τ. (ii) σ|ρ. (iii) There exists an a ∈ k^{∗} with σ ⊗ [a] < ρ. (iv) ρ ⊗ k(σ) ∼ 0.

(v) There exists a place λ: k(ρ) → k(σ) ∪ ∞.

Proof. The implications (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) and (iv) ⇔ (v) are obvious.

(iv) ⇒ (iii): Let t = (t_{1}, …, t_{n}) be a tuple of n: = dim σ unknowns over k. We can treat σ(t) ∈ k[t] as a monic polynomial. By the Norm Theorem, (iv) implies that σ(t) is a similarity norm of ρ_{k(t)} and thus that σ(t) is represented byρ_{k(t)}. The Subform Theorem then gives us σ ⊗ [1] < ρ.

(iii) ⇒ (i): We have a ∈ D(ρ), hence ρ ≅ aρ, and so σ ⊗ [1] < ρ. By Theorem 4 we then get ρ ≅ σ ⊗ τ for some quadratic Pfister form τ.

Starting from Theorem 4 we would also like to establish a counterpart to Theorem 3, dealing with bilinear Pfister forms that simultaneously divide two given quadratic Pfister forms. We need a further lemma for this purpose.

Lemma 3. Let σ be a bilinear Pfister form and ρ an anisotropic quadratic Pfister form over k. Furthermore let a ≠ 0 be an element of k with

σ ⊗ [1] ⊥ [a] < ρ.

Then there exists a quadratic Pfister form τ over k such that

ρ ≅ σ ⊗ ⟨1, a⟩ ⊗ τ.

Proof. Let n: = dim σ (a 2-power), and let (t, u) = (t_{1}, …, t_{n}, u_{1}, …, u_{n}) be a tuple of 2n unknowns over k. The form σ_{k(t,u)} is round (§16, Theorem 1). Hence there exists an n-tuple v ∈ k(t, u)^{n} with

σ(t) + aσ(u) = σ(u)(σ(v) + a).

By our assumption the elements σ(u) and σ(v) + a are represented by ρ_{k(t,u)}. Since ρ_{k(t,u)} is round as well, σ(t) + aσ(u) is represented by ρ_{k(t,u)}. By the Subform Theorem we then get

σ ⊗ ⟨1, a⟩ ⊗ [1] < ρ ,

and we can conclude the proof by applying Theorem 4.

Definition 4. (a) Let r ∈ ℕ_{0}. We call two quadratic Pfister forms ρ_{1}, ρ_{2} over k bilinearly r-linked if there exist a bilinear r-fold Pfister form σ and quadratic Pfister forms τ_{1}, τ_{2} over k such that

ρ_{1} ≅ σ ⊗ τ_{1}, ρ_{2} ≅ σ ⊗ τ_{2}.

(b) We define the bilinear linkage number i(ρ_{1}, ρ_{2}) of ρ_{1} and ρ_{2} as follows: If ρ_{1} is neither divisible by ρ_{2}, nor ρ_{2} divisible by ρ_{1}, then i(ρ_{1}, ρ_{2}) is the largest number r such that ρ_{1} and ρ_{2} are bilinearly r-linked. In the cases ρ_{1}|ρ_{2} and ρ_{2}|ρ_{1} we let i(ρ_{1}, ρ_{2}) be the number d such that 2^{d} = min(dim ρ_{1}, dim ρ_{2}).

Remark. If char k ≠ 2, it follows immediately from Definitions 2 and 4 that i(ρ_{1}, ρ_{2}) coincides with the quadratic linkage number j(ρ_{1}, ρ_{2}), which we then simply call the linkage number of ρ_{1} and ρ_{2}.

Theorem 6. ([EL, Prop.4.4] for char k ≠ 2). Let ρ_{1} and ρ_{2} be two anisotropic quadratic Pfister forms over k.

(i) The Witt index ind(ρ_{1} ⊥ −ρ_{2}) is a 2-power 2^{r} with r ≥ j(ρ_{1}, ρ_{2}) and r ≥ i(ρ_{1}, ρ_{2}). If j(ρ_{1}, ρ_{2}) > 0, then r = j(ρ_{1}, ρ_{2}). Otherwise r = i(ρ_{1}, ρ_{2}).

(ii) Let σ be a bilinear s-fold Pfister form with 0 ≤ s < r that divides ρ_{1} and

ρ_{2}, i.e.

ρ_{1} ≅ σ ⊗ τ_{1}, ρ_{2} ≅ σ ⊗ τ_{2}

with quadratic Pfister forms τ_{1}, τ_{2} (cf. Theorem 2). Then there exists either an (r − s)-fold bilinear Pfister form γ or an (r − s)-fold quadratic Pfister form γ such that σ ⊗ γ divides both Pfister forms ρ_{1} and ρ_{2}, i.e. either

ρ_{1} ≅ σ ⊗ γ ⊗ μ_{1}, ρ_{2} ≅ σ ⊗ γ ⊗ μ_{2}

with quadratic Pfister forms μ_{1}, μ_{2}, or

ρ_{1} ≅ α_{1} ⊗ σ ⊗ γ, ρ_{2} ≅ α_{2} ⊗ σ ⊗ γ

with bilinear Pfister forms α_{1}, α_{2}.

Proof. We proceed as in the proof of Theorem 3 and assume that we have factorizations ρ_{1} ≅ σ ⊗ τ_{1}, ρ_{2} ≅ σ ⊗ τ_{2} with quadratic Pfister forms τ_{1}, τ_{2} and a bilinear Pfister form σ. { We allow σ = ⟨1⟩.} We now want to verify the following two statements which, together with Theorem 4, will establish the proof.

(a) ind(ρ_{1} ⊥ −ρ_{2}) ≥ dim σ.

(b) If ind(ρ_{1} ⊥ ρ_{2}) > dim σ, then either there exists an a ∈ k^{∗} and quadratic Pfister forms μ_{1}, μ_{2} such that

(1)  ρ_{1} ≅ σ ⊗ ⟨1, a⟩ ⊗ μ_{1},  ρ_{2} ≅ σ ⊗ ⟨1, a⟩ ⊗ μ_{2},

or there exist over k a quadratic Pfister form μ of dimension 2 and bilinear Pfister forms α_{1}, α_{2} such that

(2)  ρ_{1} ≅ α_{1} ⊗ σ ⊗ μ,  ρ_{2} ≅ α_{2} ⊗ σ ⊗ μ.

Let (E_{1}, q_{1}) and (E_{2}, q_{2}) be quadratic spaces associated to ρ_{1} and ρ_{2}, and let (E, q): = (E_{1}, q_{1}) ⊥ (E_{2}, −q_{2}). The corresponding vector space decomposition E = E_{1} ⊕ E_{2} gives rise to projections p_{1}: E ⟶ E_{1}, p_{2}: E ⟶ E_{2}.

We have [1] < τ_{1} and [1] < τ_{2}, and thus

σ ⊗ [1, −1] < σ ⊗ τ_{1} ⊥ σ ⊗ (−τ_{2}) ≅ ρ_{1} ⊥ −ρ_{2}.

Hence E contains a subspace U with q(U) = 0, dim U = dim σ, such that the form σ ⊗ [1] corresponds to p_{1}(U) as a subspace of (E_{1}, q_{1}) as well as to p_{2}(U) as a subspace of (E_{2}, q_{2}). To see this, note that p_{i} (i = 1, 2) maps the vector space U linearly isomorphic to p_{i}(U) since E_{1} and E_{2} are anisotropic as subspaces of (E, q).

Since E contains the totally isotropic subspace U, it is already clear that ind(q_{1} ⊥ −q_{2}) ≥ dim σ.

Assume now that ind(q_{1} ⊥ −q_{2}) > dim σ. Then (E, q) contains a totally isotropic subspace V = U + kv of dimension 1 + dim σ. Again we know that p_{i} maps the vector space V linearly isomorphic to p_{i}(V) (i = 1, 2), and for arbitrary x ∈ V we have:

q_{1}(p_{1}(x)) = q_{2}(p_{2}(x)),

and thus also for x, y ∈ V:

B_{1}(p_{1}(x), p_{1}(y)) = B_{2}(p_{2}(x), p_{2}(y)),

where B_{i} denotes the bilinear form B_{q_{i}} corresponding to q_{i}. We distinguish two cases:

Case 1: B_{1}(p_{1}(v), p_{1}(u)) = 0. Now σ ⊥ [a] corresponds to p_{1}(V) with respect to q_{1} and also to p_{2}(V) with respect to q_{2} where a: = q_{1}(p_{1}(v)) = q_{2}(p_{2}(v)). Thus σ ⊥ [a] is a subform of both ρ_{1} and ρ_{2}. By Lemma 3 we obtain (1) with quadratic Pfister forms μ_{1} and μ_{2}.

Case 2: B_{1}(p_{1}(v), p_{1}(u)) ≠ 0. We choose an orthogonal basis e_{1}, …, e_{n} of p_{1}(U) with respect to q_{1} with B_{1}(e_{1}, p_{1}(v)) = 1, B_{1}(e_{i}, p_{1}(v)) = 0 for 2 ≤ i ≤ n, which is certainly possible. Then, corresponding to p_{1}(V) with respect to q_{1} and also to p_{2}(V) with respect to q_{2}, we have the form

[{a_{1} \atop 1} {1 \atop b_{1}} ] ⊥ [a_{2}, …, a_{n}]

with b_{1}: = q_{1}(p_{1}(v)) and a_{i} = q_{1}(e_{i}), and we have

σ ⊗ [1] ≅ [a_{1}, a_{2}, …, a_{n}].

Now by §6, Lemma 2,

[{a_{1} \atop 1} {1 \atop b_{1}} ] ≅ ⟨a_{1}⟩ ⊗ [{1 \atop 1} {1 \atop b}]

with b: = a_{1}b_{1}. Since the three forms σ, ρ_{1}, ρ_{2} are round, it follows that

[{1 \atop 1} {1 \atop b}] ⊥ [a_{2}a_{1}^{−1}, …, a_{n}a_{1}^{−1}] < ρ_{1}

for i = 1 and i = 2, and also that

[1, a_{2}a_{1}^{−1}, …, a_{n}a_{1}^{−1}] ≅ σ ⊗ [1].

Hence

[{1 \atop 1} {1 \atop b}] ⊥ σ′ ⊗ [1] < ρ_{i}

for i = 1 and i = 2. Now Lemma 2 tells us that the factorizations (2) above hold with μ = [{1 \atop 1} {1 \atop b}] and bilinear Pfister forms α_{1} and α_{2}.

Problem. Let char k = 2, a ∈ k^{∗}, and let ρ_{1}, ρ_{2} be 2-fold anisotropic quadratic Pfister forms with ⟨1, a⟩|ρ_{1} and ⟨1, a⟩|ρ_{2}. Does there always exist a quadratic form τ = [{1 \atop 1} {1 \atop b}] over k with τ|ρ_{1} and τ|ρ_{2}?

The author has not yet been able to give either a positive or a negative answer to this question. If the answer is “yes”, then by Theorem 6 it is immediate that for two anisotropic quadratic Pfister forms ρ_{1}, ρ_{2} over k we always have that

ind(ρ_{1} ⊥ −ρ_{2}) = 2^{j(ρ_{1},ρ_{2})}.

Finally we remark that R. Baeza proved several theorems in this section, or parts of them, more generally over semi-local rings instead of fields, see in particular [Ba_{1}, Chap.IV, §4]. As our proofs rely on the Norm Theorem, the Subform Theorem, and elements of generic splitting, it is till this day not possible to preserve them over semi-local rings. Thus Baeza’s proofs differ from the ones we presented here. They are substantially more “constructive” than ours. Therefore they deserve also over fields an interest for their own sake, just as the proofs of Elman and Lam in [EL] when the characteristic is different from two.

§22 Pfister Neighbours and Excellent Forms

In the following k denotes a field of arbitrary characteristic and a “form” will always mean a nondegenerate quadratic form.

Definition 1. (cf. [K_{6}, Def.7.4] for char k ≠ 2). A form φ over k is called a Pfister neighbour if there exist a quadratic Pfister form τ over k and an element a ∈ k^{∗} such that φ < aτ and dim φ > \frac{1}{2} dim τ. More precisely we then call φ a neighbour of the Pfister form τ.

In §16 we already introduced the notion of close Pfister neighbour for the case char k = 2. Close Pfister neighbours are of course Pfister neighbours in the current sense.

If φ is a neighbour of a Pfister form τ, then we must clearly have dim φ ≥ 2, and in case dim φ = 2 we must moreover have φ ≅ aτ.

Theorem 1. If φ is a neighbour of a Pfister form τ and a is an element of k^{∗} with φ < aτ, then the form τ and also the form aτ are up to isometry uniquely determined by φ. The fields k(φ) and k(τ) are specialization equivalent over k.

Proof. The form aτ_{K} is hyperbolic over the field K: = k(τ). Let E = (E, q) be a quadratic space corresponding to aτ_{K} and F = (F, q|F) a quadratic subspace of E corresponding to the subform φ_{K} of aτ_{K}. Let dim E = 2^{d}. There exists a vector subspace U of E of dimension 2^{d−1} with q(U) = 0. From the obvious inequality

dim(F ∩ U) + dim E ≥ dim F + dim U

and the assumption that dim F > 2^{d−1} one sees that dim(F ∩ U) > 0, and thus that F ∩U ≠ {0}. Hence F is isotropic and thus the form φ_{K} = φ⊗k(τ) is isotropic. Therefore there exists a place from k(φ) to k(τ) over k. Since φ is a subform of aτ, the form τ ⊗ k(φ) is also isotropic. Thus there also exists a place from k(τ) to k(φ). (A conclusion we already reached many times!) Hence k(τ) ∼_{k} k(φ).

By the corollary of §21, Theorem 2 it now follows that τ is uniquely determined by φ up to isometry. If φ < aτ and c ∈ D(φ) is arbitrarily chosen, then ac ∈ D(τ), thus τ ≅ acτ, and so aτ ≅ cτ. This shows that also the form aτ is uniquely determined by φ up to isometry.

Definition 2. Let φ be a Pfister neighbour, thus φ < aτ where τ is a Pfister form and dim φ > \frac{1}{2} dim τ. Then we call the by φ up to isometry uniquely determined polar Pol_{aτ} (φ) of φ in aτ (cf. §16) the complementary form of φ. Furthermore we call τ the neighbouring Pfister form of φ.

Remarks. (1) Again let 2^{d} = dim τ. Clearly the complementary form η = Pol_{aτ} (φ) of φ has dimension 2^{d} − dim φ < 2^{d−1}. Furthermore φ and η have the same quasi-linear part, QL(φ) = QL(η), as already observed in §16, Lemma 1.

(2) In particular η is strictly regular if and only if φ is strictly regular. We then have φ ⊥ η ≅ aτ. This is always the case when k has characteristic ≠ 2. In characteristic 2 we also have that η is regular if and only if φ is regular.

(3) By §16, φ is a close Pfister neighbour if and only if η is quasi-linear. We then have η = QL(φ). Moreover, if φ is anisotropic, then φ has height 1 by §16. If φ is isotropic, then φ splits.

Theorem 2. Let φ be an anisotropic Pfister neighbour over k with complementary form η. Then φ ⊗ k(φ) ∼ (−η) ⊗ k(φ).

Proof. Let τ be the neighbouring Pfister form of φ and φ < aτ. If φ is strictly regular, then aτ ≅ φ ⊥ η and τ ⊗ k(φ) ∼ 0, and thus certainly φ ⊗ k(φ) ∼ (−η) ⊗ k(φ). This holds in particular if k is of characteristic ≠ 2.

Assume now that char k = 2 and that φ is not strictly regular.^{36} Let χ be the quasi-linear part of φ, χ = QL(φ). By the scholium in §16 we have an orthogonal decomposition τ = μ ⊥ γ ⊥ ρ with

χ < μ, dim μ = 2 dim χ, φ ≅ χ ⊥ γ, η = χ ⊥ ρ.

With K: = k(φ) we obtain

μ_{K} ⊥ γ_{K} ⊥ ρ_{K} ∼ 0,

thus γ_{K} ∼ μ_{K} ⊥ ρ_{K} (note that char k = 2!), and so

φ_{K} ∼ χ_{K} ⊥ μ_{K} ⊥ ρ_{K}.

Let χ ≅ [a_{1}, …, a_{r}]. Then

μ_{K} ≅ [{a_{1} \atop 1} {1 \atop b_{1}} ] ⊥ ⋯ ⊥ [{a_{r} \atop 1} {1 \atop b_{r}} ]

with elements b_{i} of k. Hence

χ ⊥ μ ≅ χ ⊥ [{0 \atop 1} {1 \atop b_{1}} ] ⊥ ⋯ ⊥ [{0 \atop 1} {1 \atop b_{r}} ] ≅ χ ⊥ r × H,

and we obtain

φ_{K} ∼ χ_{K} ⊥ ρ_{K} = η_{K}.

We return to the theory of generic splitting established in §12.

36The following calculation generalizes the proof of Theorem 5, (i) – (iii) in §16.

For a given form φ, even of small dimension, it is in general very complicated to explicitly determine the splitting pattern SP(φ) or even the higher kernel forms. However, we will now specify a larger class of forms for which it is possible to do this, the “excellent forms”. We define a class of forms by induction on the dimension (cf. [K_{6}, §7] for char k ≠ 2).

Definition 3. A form φ is called excellent if dim φ ≤ 1, or φ is quasi-linear, or φ is a Pfister neighbour with excellent complementary form.

{Note that if the ground field k has characteristic 2 and if dim φ ≤ 1, then φ is of course quasi-linear. In characteristic ≠ 2 on the other hand, there are no quasi-linear forms φ ≠ 0.}

Thus, by this definition, a form φ over k is excellent if and only if there exists a finite sequence of forms

η_{0} = φ, η_{1}, …, η_{t}

over k such that η_{t} is quasi-linear or at most one-dimensional, and every form η_{r} with 0 ≤ r < t is a Pfister neighbour with complementary form η_{r+1}. We call the forms η_{r} the higher complementary forms of φ, more specifically we call η_{r} the r-th complementary form of φ (0 ≤ r ≤ t).

Theorem 3. Let φ be an anisotropic excellent form over k and let (η_{i} | 0 ≤ i ≤ t) be the sequence of higher complementary forms of φ. For every r ∈ {0, 1, …, t}, let K_{r} denote the free composite of the fields k(η_{i}) with 0 ≤ i < r over k. {Read K_{0} = k for r = 0.}

Claim: (K_{r} | 0 ≤ r ≤ t) is a generic splitting tower of φ. For every r ∈ {0, 1, …, t}, φ ⊗ K_{r} has kernel form (−1)^{r}η_{r} ⊗ K_{r}. In particular, φ has height t.

Proof. By induction on r. For r = 0 we don’t have to show anything, so suppose that r > 0. Let τ_{0} and τ_{1} be the neighbouring Pfister forms of φ = η_{0} and η_{1} respectively. Thus there are elements a_{0}, a_{1} in k^{∗} with φ < a_{0}τ_{0}, η_{1} < a_{1}τ_{1}, η_{1} = Pol_{a_{0}τ_{0}}(φ). Furthermore, K_{1} = k(φ). By Theorem 2 we have φ ⊗ K_{1} ∼ (−η_{1}) ⊗ K_{1}, and by Theorem 1 we have k(φ) ∼_{k} k(τ_{0}), k(η_{1}) ∼_{k} k(τ_{1}).

If η_{1} ⊗ K_{1} were isotropic, then τ_{1} ⊗ k(τ_{0}) would also be isotropic, and thus hyperbolic. This implies, by §21, Theorem 1 (or §21, Theorem 2), that the form τ_{1} is divisible by τ_{0}. However, this is absurd since τ_{1} has smaller dimension than τ_{0}. Hence η_{1} ⊗ K_{1} is anisotropic. Thus (−η_{1}) ⊗ K_{1} is the kernel form of φ ⊗ K_{1}.

By definition of excellent forms it is clear that η_{1} ⊗ K_{1} is excellent and that for every r ∈ {1, …, t} the form η_{r} ⊗K_{1} is the (r −1)-st complementary form of η_{1} ⊗ K_{1}. If we now apply the induction hypothesis to η_{1} ⊗ K_{1} we obtain the complete statement of the theorem. With regards to this, one should keep in mind that K_{1}(η_{i} ⊗ K_{1}) is the free composite of K_{1} and k(η_{i}) over k (1 ≤ i ≤ t) and consequently that K_{r} is the free composite of the fields K_{1}(η_{i} ⊗ K_{i}) with 1 ≤ i < r over K_{1} (1 ≤ r ≤ t).

In which dimensions do there exist, for a given field k, anisotropic excellent forms over k? We want to discuss this question in the (difficult) case when k has characteristic 2.

Scholium. Let char k = 2. If φ is an anisotropic excellent form of dimension n over k, then the neighbouring Pfister form τ of φ is again anisotropic, and dim τ is clearly the smallest 2-power 2^{d} ≥ n.

Conversely, let τ be an anisotropic Pfister form. In this case we construct an excellent form with neighbouring form τ as follows: We choose a sequence (τ_{r} | 1 ≤ r ≤ t) of Pfister subforms of τ with τ_{1} = τ, τ_{r} | τ_{r−1} for 2 ≤ r ≤ t, and a quasi-linear subform η of τ_{t} with dim η < \frac{1}{2} dim τ_{t}, which is always possible. {t ≥ 1. It is allowed that η = 0.} Then we define successive forms η_{t}, η_{t−1}, …, η_{0} in such a way that

η_{t} = η , η_{r} < τ_{r} , η_{r−1} < τ_{r} , η_{r−1} = Pol_{τ_{r}}(η_{r}) for 1 ≤ r ≤ t.

Now φ: = η_{0} is an excellent form with neighbour τ and higher complementary forms η_{r} (0 ≤ r ≤ t). The form φ is anisotropic and has quasi-linear part η. Up to a scalar factor one can obtain every anisotropic excellent form over k in this manner.

Next we show that over suitable fields of characteristic 2 there exist anisotropic Pfister forms of arbitrarily high dimension.

Theorem 4. Let κ be any field and k = κ(u, u_{1}, …, u_{n}), where u, u_{1}, …, u_{n} is a tuple of unknowns. The Pfister form ⟨1, u_{1}⟩ ⊗ ⋯ ⊗ ⟨1, u_{n}⟩ ⊗ [{1 \atop 1} {1 \atop u}] over k is anisotropic.

Proof. We assume that κ has characteristic 2 and leave the proof for the case char κ ≠ 2 to the Reader. We work over the iterated power series field

K: = κ((u))((u_{1})) … ((u_{n})),

which contains k. Let τ be the Pfister form from above, considered as a form over K. We show by induction on n that τ is anisotropic.

n = 0 : The equation u = x^{2} − x is with x ∈ κ((u)) unsolvable, so that the form [{1 \atop 1} {1 \atop u}] is anisotropic over κ((u)).

n − 1 → n : Let F: = κ((u))((u_{1})) … ((u_{n−1})). We have τ = ρ_{K} ⊥ u_{n}ρ_{K} with ρ an anisotropic Pfister form over F by the induction hypothesis. There is a place λ: K → F ∪ ∞ over F belonging to the obviously discrete valuation of K over F with prime element u_{n}. The form ρ_{K} has good reduction with respect to λ and λ_{∗}(ρ_{K}) = ρ. We further have that λ(u_{n}c^{2}) = 0 or = ∞ for every c ∈ K^{∗}. If τ were isotropic, and thus hyperbolic, then we would have ρ_{K} ≅ u_{n}ρ_{K}. Applying λ_{W} would then give ρ ∼ 0, a contradiction. Therefore τ is anisotropic.

Example. We choose two natural numbers n > m, and a sequence of natural numbers

s_{1} = n > s_{2} > ⋯ > s_{t} = m.

Let k = κ(u, u_{1}, …, u_{n}), as in Theorem 4, and char κ = 2. We form the – by Theorem 4 anisotropic – Pfister forms

τ_{r}: = ⟨1, u_{1}⟩ ⊗ ⋯ ⊗ ⟨1, u_{s_{r}}⟩ ⊗ [{1 \atop 1} {1 \atop u}]

(1 ≤ r ≤ t), and then the quasi-linear form

η: = [1, u_{1}, u_{2}, …, u_{s}]

for some s < s_{t} = m. We have η < τ_{t} and dim η < \frac{1}{2} dim τ_{t}. Using the procedure outlined in the scholium above we obtain an anisotropic excellent form φ of height t over k with (s + 1)-dimensional quasi-linear part η.

We utilize Theorem 4 and this example to illustrate principles about the specialization of quadratic forms, obtained earlier (§15, Theorem 5, §11, Theorem 2). We consider the question of when an excellent form has good reduction with respect to a place, and its specialization is again excellent.

Theorem 4 gives us the “generic” (n + 1)-fold Pfister form of a field κ. Namely, if ρ is an (n + 1)-fold Pfister form over κ, and if we have chosen a representation

ρ ≅ ⟨1, a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, a_{n}⟩ ⊗ [{1 \atop 1} {1 \atop a}]

(a ∈ κ, all a_{i} ∈ κ), then there exists a place λ: κ(u, u_{1}, …, u_{n}) → κ ∪ ∞ over κ with λ(u) = a, λ(u_{i}) = a_{i} for i = 1, …, n. The form

τ: = ⟨1, u_{1}⟩ ⊗ ⋯ ⊗ ⟨1, u_{n}⟩ ⊗ [{1 \atop 1} {1 \atop u}]

has good reduction with respect to every such place λ, and λ_{∗}(τ) ≅ ρ.

Now suppose that ρ is anisotropic. We choose a binary quadratic subform γ of τ with 1 ∈ D(γ), which has good reduction with respect to λ, e.g. γ = [{1 \atop 1} {1 \atop u}]. If σ is a Pfister form over k with γ < σ < τ, then σ is clearly obedient with respect to λ. By §15, Theorem 5, and §11, Theorem 2, it follows that σ has good reduction with respect to λ and that λ_{∗}(σ) is a subform of ρ = λ_{∗}(τ). Furthermore, λ_{∗}(γ) is a subform of λ_{∗}(σ).

We want to obtain that λ_{∗}(σ) is again a Pfister form. By §12, λ_{∗}(σ) has height ≤ 1. Since λ_{∗}(σ) is anisotropic, and thus doesn’t split, we must have h(λ_{∗}(σ)) = 1. Furthermore, [1] < σ implies, again by §15, Theorem 5, etc. that [1] = λ_{∗}([1]) is a subform of λ_{∗}(σ), thus 1 ∈ D(λ_{∗}(σ)). Finally §20, Theorem 5, tells us that λ_{∗}(σ) is a Pfister form. It divides λ_{∗}(τ) = ρ.

We turn our attention to the construction of excellent forms in the scholium above. There we choose the Pfister form τ_{t} so that γ < τ_{t}. Now all forms τ_{r} have good reduction with respect to λ, every λ_{∗}(τ_{r}) is a Pfister form, and λ_{∗}(τ_{r}) | λ_{∗}(τ_{r−1}) for 2 ≤ r ≤ t. The quasi-linear form η is automatically obedient with respect to λ. Thus, by §15, Theorem 5, etc., it has good reduction with respect to λ and λ_{∗}(η) is a subform of λ_{∗}(τ_{t}).

If the complementary form η_{t−1} of η = η_{t} in τ_{t} is obedient with respect to λ, then we obtain in the same fashion that η_{t−1} has good reduction with respect to λ, and that λ_{∗}(η_{t−1}) < λ_{∗}(τ_{t}). Furthermore, λ_{∗}(η_{t−1}) is now orthogonal to λ_{∗}(η_{t}), and by dimension considerations it is now clear that λ_{∗}(η_{t−1}) is the polar of λ_{∗}(η_{t}) in λ_{∗}(τ_{t}).

We can continue this line of reasoning in so far that it is guaranteed that all η_{r} are obedient with respect to λ, and obtain finally that the excellent form φ = η_{0} has good reduction with respect to λ, and that λ_{∗}(φ) is an excellent subform of λ_{∗}(τ) = ρ with ρ as neighbour.

If one considers γ = [{1 \atop 1} {1 \atop u}], and starts with η = [1, u_{1}, u_{2}, …, u_{s}], as in the example above, one can reason inductively that all η_{r} are indeed obedient with respect to λ, and thus actually have good reduction with respect to λ.

Now, if ρ is anisotropic, then it follows from §15, Theorem 5, that every subform σ of τ has good reduction with respect to λ, and that λ_{∗}(σ) is a subform of ρ. If σ is a Pfister form, then λ_{∗}(σ) has height ≤ 1 by §12. Since λ_{∗}(σ) is anisotropic, and so doesn’t split, λ_{∗}(σ) must have height exactly 1. Moreover, λ_{∗}(σ) is strictly regular, since this is the case for σ.

Furthermore, [1] < σ implies that [1] = λ_{∗}[1] is a subform of λ_{∗}(σ). Now we conclude with §20, Theorem 5, that λ_{∗}(σ) is a Pfister form (and it divides the form ρ since it is a subform of ρ).

Every Pfister subform σ of τ thus has good reduction with respect to λ, and provides a Pfister subform λ_{∗}(σ) of ρ. By the scholium above it is now clear that the anisotropic excellent form φ in our example above has good reduction with respect to λ, and that λ_{∗}(φ) is a neighbour of the Pfister form ρ.

Nevertheless we cannot consider φ to be a “generic” excellent form of height 1 with particular splitting pattern for κ since there are too many other possibilities to construct excellent subforms of τ. Our reasoning about the specialization of excellent forms remains valid in the general situation however. We record

Theorem 5. Let λ: K → L ∪ ∞ be a place and τ a Pfister form which has good reduction with respect to λ. Assume that the form λ_{∗}(τ) is anisotropic. Then λ_{∗}(τ) is again a Pfister form. Every excellent subform φ of τ which is a neighbour of τ has good reduction with respect to λ, and λ_{∗}(φ) is an excellent form, neighbouring λ_{∗}(τ), of the same height as φ.

In the situation of the last theorem, the form λ_{∗}(τ) is furthermore also a Pfister form when it is isotropic, since it follows already from the generic splitting theory in §9 that λ_{∗}(τ) ≅ 2^{d−1} × H with 2^{d} = dim τ.

§23 Regular Forms of Height 1

In the following let φ be a regular anisotropic quadratic form of height 1 over a field k. The purpose of this section is the proof of the following theorem.

Theorem 1. φ is excellent.

More particularly: if φ is of even dimension, then φ = aτ with a ∈ k^{∗} and τ a Pfister form. If φ is of odd dimension, then φ = a Pol_{τ} ([1]) with a ∈ k^{∗} and τ a Pfister form of dimension ≥ 4. Is furthermore char k ≠ 2, then Pol_{τ} ([1]) is the pure part of τ, cf. §16.

For φ of even dimension we already proved the theorem in §20, Theorem 5. Thus, suppose from now on that the dimension of φ is odd.

We start with the proof for char k ≠ 2 (cf. [K_{5}, p.81 ff.]). We interpret φ as a symmetric bilinear form. Upon multiplying φ by a scalar factor we may assume without loss of generality that d(φ) = 1. {The signed determinant d(φ) was introduced in §2.}

We first show that φ does not represent the element 1. By way of contradiction assume that φ does represent 1. Then we have an orthogonal decomposition φ = ⟨1⟩ ⊥ χ. Since h(φ) = 1 we have φ ⊗ k(φ) ∼ ⟨1⟩, and so χ ⊗ k(φ) ∼ 0. As usual the Norm Theorem and the Subform Theorem tell us that aφ < χ for a suitable a ∈ k^{∗}. This is absurd however since the dimension of χ is smaller than the dimension of φ. Therefore 1 ∉ D(φ). The form τ: = ⟨1⟩ ⊥ (−φ) is thus anisotropic.

Let L/k be an arbitrary field extension of k. If φ ⊗ L is anisotropic, then h(φ ⊗ L) = 1 since h(φ ⊗ L) ≤ h(φ) and h(φ ⊗ L) ≠ 0. If we apply what we just proved to φ ⊗ L instead of φ, we see that τ ⊗ L is anisotropic. On the other hand, if φ ⊗ L is isotropic, then φ ⊗ L ∼ ⟨1⟩, and so τ ⊗ L ∼ 0. This shows that τ ⊗ L is either anisotropic or hyperbolic. Thus τ has height 1. Furthermore 1 ∈ D(τ). As already established before (§20, Theorem 5), τ is a Pfister form. It follows that φ = −τ′.

Let us now continue by proving Theorem 1 for char k = 2 and dim φ odd, dim φ = 2m + 1 with m > 0. Without loss of generality we may assume that φ has quasi-linear part QL(φ) = [1]. We have to show that φ is a close neighbour (cf. §16) of a Pfister form.

We deal with the case m = 1 separately. In this case we have

φ ≅ [1] ⊥ c [{1 \atop 1} {1 \atop b}]

with c ∈ k^{∗}, b ∈ k, and so φ is a close neighbour of the Pfister form ⟨1, c⟩ ⊗ [{1 \atop 1} {1 \atop b}].

From now on let m ≥ 2, and thus dim φ ≥ 5. We require an easy lemma of a general nature that we will prove at a later stage.

Lemma. Let ψ and χ be quadratic forms over k, and assume that ψ ⊥ χ is nondegenerate. Then

ind ψ + dim χ ≥ ind(ψ ⊥ χ).

We have an orthogonal decomposition

φ ≅ [1] ⊥ ρ

with ρ strictly regular, dim ρ = 2m. Since the height of φ is 1 we have ind(φ ⊗ k(φ)) = m. The lemma implies that

ind(ρ ⊗ k(φ)) ≥ m − 1 > 0.

So, ρ ⊗ k(φ) is in particular isotropic. Since φ ⊗ k(ρ) is trivially also isotropic it follows that k(φ) ∼_{k} k(ρ) and then

ind(ρ ⊗ k(ρ)) ≥ m − 1.

Let us assume for the sake of contradiction that ind(ρ ⊗ k(ρ)) > m − 1, thus ρ ⊗ k(ρ) ∼ 0. By earlier work we have that ρ = aσ where σ is a Pfister form. Hence φ = [1] ⊥ aσ is a neighbour of the Pfister form τ: = ⟨1, a⟩ ⊗ σ. It follows from §22, Theorem 1 that k(φ) ∼_{k} k(τ). This implies k(ρ) ∼_{k} k(τ), since k(σ) is also specialization equivalent to k(φ) over k. By §21, Corollary to Theorem 1, this means σ ≅ τ, which is not possible since dim τ = 2 dim σ. Therefore

ind(ρ ⊗ k(ρ)) = m − 1.

Thus ρ ⊗ k(ρ) is the orthogonal sum of m − 1 hyperbolic planes and an anisotropic binary form. Hence ρ ⊗ k(ρ) has non-vanishing Arf invariant. Thus we certainly have Arf(ρ) ≠ 0. Let Arf(ρ) = a + ℘k and

σ: = [{1 \atop 1} {1 \atop a}] .

Consider the form τ: = σ ⊥ ρ. Then φ < τ and Arf(τ) = 0. {Note that we are now in a situation which is similar to the proof involving the form τ in char k ≠ 2.} Assume that φ⊗k(σ) is anisotropic. This form then has height 1 and

φ ⊗ k(σ) = [1] ⊥ ρ ⊗ k(σ).

Thus, we should get Arf(ρ ⊗ k(σ)) ≠ 0, which is false. Therefore φ ⊗ k(σ) is isotropic and it follows that

φ ⊗ k(σ) ≅ [1] ⊥ m × H.

§23 Regular Forms of Height 1 171

Hence, m × H < τ ⊗ k(σ). Since dim τ = 2m + 2 and Arf(τ ⊗ k(σ)) = 0 we obtain τ ⊗ k(σ) ∼ 0 and so ρ ⊗ k(σ) ∼ 0. By §21, Theorem 1, there exist a_{1}, …, a_{m} ∈ k^{∗} with

ρ ≅ ⟨a_{1}, …, a_{m}⟩ ⊗ σ.

Consequently,

τ ≅ ⟨1, a_{1}, …, a_{m}⟩ ⊗ σ.

We will now show that τ is anisotropic. Assume for the sake of contradiction that this is not so. Then there exist vectors x_{0}, x_{1}, …, x_{m} ∈ k^{2} with

σ(x_{0}) + ∑_{i=1}^{m} a_{i}σ(x_{i}) = 0

and not all x_{i} = 0. Since ρ is anisotropic, we must have x_{0} ≠ 0 and thus also σ(x_{0}) ≠ 0. Thus there are vectors y_{1}, …, y_{m} ∈ k^{2} with σ(y_{i}) = σ(x_{0})^{−1}σ(x_{i}) for 1 ≤ i ≤ m. Hence,

1 + ∑_{i=1}^{m} a_{i}σ(y_{i}) = 0.

This contradicts the anisotropy of φ. Thus τ is indeed anisotropic.

Now it follows easily that φ ⊗ k(τ) is isotropic. Indeed, if φ ⊗ k(τ) were anisotropic, then we could apply what we just proved to the form φ ⊗ k(τ) of height 1 and we would conclude that τ ⊗ k(τ) would be anisotropic, which is of course false.

Hence there exists a place from k(φ) to k(τ) over k, and it follows that

φ ⊗ k(τ) ≅ [1] ⊥ m × H.

Thus τ ⊗ k(τ) > m × H and, since Arf(τ) = 0 and dim τ = 2m + 2, even τ ⊗ k(τ) ∼ 0. Furthermore, 1 ∈ D(τ) and τ is anisotropic. By earlier work τ is a Pfister form. The form φ is a close neighbour of this Pfister form.

Let us finally supply the proof of the lemma above. Then our theorem will be completely proved.

The field k may have arbitrary characteristic. Let ψ and χ be quadratic forms over k. If ind(ψ ⊥ χ) ≤ dim χ, there is nothing to prove. Thus, suppose ind(ψ ⊥ χ) > dim χ. Let E be the quadratic space of ψ and G the quadratic space of χ. The space E ⊥ G, nondegenerate by assumption, contains a totally isotropic subspace V with dim V = ind(ψ ⊥ χ). Let π: V → G be the restriction of the natural projection E ⊥ G → G to V. The kernel of π is then V ∩ E. This space is totally isotropic and

dim(V ∩ E) + dim G ≥ dim V.

Therefore,

dim(V ∩ E) ≥ ind(ψ ⊥ χ) − dim χ,

which establishes the assertion of the lemma.

172 Chapter III. Some Applications

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§24 Some Open Problems in Characteristic 2

We subdivide this section into parts (A) – (D), posing four problems which seem to be open.

As before, the word “form” stands for nondegenerate quadratic form and the ground field k is arbitrary, unless indicated otherwise.

(A) Let char k = 2. Which nonregular forms of height 1 are there?

If φ is anisotropic and dim φ = 2+dim QL(φ), then φ surely has height 1. Not every such form is excellent! As a simple example we take an anisotropic 4-dimensional form

φ = [a_{1}, a_{2}] ⊥ [{a_{3} \atop 1} {1 \atop b_{3}} ] .

If φ were excellent, it would have to be a Pfister form up to a scalar factor, which is not the case.

We give another example. Let

φ = [a_{1}, a_{2}, a_{3}, a_{4}] ⊥ [{a_{5} \atop 1} {1 \atop b_{5}} ]

be anisotropic. {E.g., let a_{1}, …, a_{5}, b_{5} be unknowns over a field κ and let k = κ(a_{1}, …, a_{5}, b_{5}).} We have h(φ) = 1 and dim φ = 6. Let φ < τ and τ a strictly regular hull of φ (cf. §16). Let χ: = QL(φ). By the Scholium in §16 we have

φ = χ ⊥ γ, τ = μ ⊥ γ

with χ < μ, dim μ = 2 dim χ = 8. Thus, dim τ = 8 + 2 = 10. The form φ is certainly not a Pfister neighbour.

Let us ask a more precise question: Do there exist any anisotropic forms of height 1, other than close Pfister neighbours and anisotropic forms φ with dim(φ) = 2 + dim QL(φ)?

(B) Let φ be a neighbour of a Pfister form τ and let η be the complementary form. Then φ⊗k(φ) ∼ (−η)⊗k(φ) (§20, Theorem 2). Is η⊗k(φ) anisotropic?

If the characteristic of the ground field is different from two, this is true, as shown originally by R. Fitzgerald [F]. Hence (−η)⊗k(φ) is the first higher kernel form of φ. Later on, D. Hoffmann proved the following far-reaching theorem:

Theorem 1 [H_{4}]. Let char k ≠ 2 and let φ and η be anisotropic forms over k. Assume that there exists a 2-power 2^{m} such that dim η ≤ 2^{m} < dim φ. Then η ⊗ k(φ) is anisotropic.

Does this statement remain true in characteristic 2?^{37}

One could simply try to extend the proofs of Fitzgerald and Hoffmann to characteristic two. This approach, however, leads to problems very soon since these proofs make extensive use of the following argument: let ψ_{1} and ψ_{2} be anisotropic forms over k whose orthogonal sum ψ_{1} ⊥ ψ_{2} is isotropic. Then there exists an a ∈ k^{∗} such that ψ_{1} = [a] ⊥ ψ′_{1} and ψ_{2} = [−a] ⊥ ψ′_{2}. But this is only correct when k has characteristic ≠ 2.^{38} In characteristic 2 there does not seem to exist anything that achieves results similar to this simple, but effective argument.

(C) Definition. Let L/k be a field extension and ψ a form over L. We say that ψ is definable over k if there exists a form η over k such that ψ ≅ η ⊗ L.

Let φ be a form over k. Let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower of φ and let φ_{r}: = ker(φ ⊗ K_{r}), the r-th higher kernel form of φ (0 ≤ r ≤ h). We are interested in the situation where all φ_{r} are definable over k. One should observe that this property is independent of the choice of the generic splitting tower (K_{r} | 0 ≤ r ≤ h).

Problem. For which anisotropic forms φ over k are all higher kernel forms of φ definable over k?

Theorem 3 from §22 tells us that this is the case for excellent forms. For char k ≠ 2 it is known that there are no other forms with this property [K_{6}, Th.7.14]. {For this reason many texts, e.g., also [KS, §13], define excellence via this property when char k ≠ 2. We will not do that here.}

Obviously every anisotropic form of height 1 satisfies this property. Among such forms there are non-excellent ones, as seen under (A). In case the first problem under (B) has an affirmative answer,^{35} it is not so hard to obtain further examples of anisotropic forms whose higher kernel forms are definable over the ground field. Simply embed η into an anisotropic Pfister form τ, of dimension > 2 dim η (we may assume without loss of generality that 1 ∈ D(η)). Then all higher kernel forms of the polar φ of η in τ are also definable over k.

(D) Let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower of a form φ over k and let (φ_{r} | 0 ≤ r ≤ h) be the associated higher kernel forms of φ. We fix a number r with 0 < r < h. We assume that φ_{r} is definable over k. Is there, up to isometry, only one form η over k such that φ_{r} ≅ η ⊗ K_{r}? {Note that this is trivially true for r = 0 and r = h.}

37See Addendum at the end of this section! 38In characteristic 2 the argument remains valid only for symmetric bilinear forms.

§24 Some Open Problems in Characteristic 2 175

It is known that the answer is “yes” for char k ≠ 2. A proof can be found in [K_{6}, §7]. Fortunately this proof remains valid in characteristic 2 when the form φ is strictly regular. We will reproduce it below. In slightly more generality we can show:

Theorem 2. Let φ be strictly regular. Let η_{1} and η_{2} be anisotropic forms over k with η_{1} ⊗ K_{r} ∼ η_{2} ⊗ K_{r} ∼ φ ⊗ K_{r} and dim η_{1} < dim φ_{r−1}, dim η_{2} < dim φ_{r−1}. Then η_{1} ≅ η_{2}.

Proof [K_{6}, p.1 ff.]. We assume without loss of generality that the form η_{1} has minimal dimension among all anisotropic forms δ over k with δ⊗K_{r} ∼ φ⊗K_{r}.

We first settle the case r = 1. Assume that η_{1} and η_{2} are not isometric. Then η_{1} ⊥ (−η_{2}) has a kernel form ζ ≠ 0 with ζ ⊗ k(φ) ∼ 0. By the Norm Theorem and the Subform Theorem there exist a form ψ over k and an element a ∈ k^{∗} such that

φ ⊥ ψ ≅ aζ.

It follows that φ ⊗ k(φ) ∼ (−ψ) ⊗ k(φ). Now

dim η_{1} + dim η_{2} ≥ dim ζ = dim φ + dim ψ,

and so

dim η_{1} − dim ψ ≥ dim φ − dim η_{2} > 0.

This contradicts the minimality of dim η_{1}. Therefore, η_{1} ≅ η_{2}.

Assume next that r ≥ 2. We proceed by induction on r. Assume that η_{1} and η_{2} are not isometric, and thus η_{1} ≁ η_{2}. Let s be the largest integer between 0 and r − 1 such that η_{1} ⊗ K_{s} ≁ η_{2} ⊗ K_{s}. If s > 0 we apply the induction hypothesis for r −s to the kernel forms of φ⊗K_{s}, η_{1} ⊗K_{s}, η_{2} ⊗K_{s}, and obtain a contradiction. Hence s = 0. For the kernel form ζ of η_{1} ⊥ (−η_{2}) we have again that ζ ⊗ k(φ) ∼ 0 and, just as in the case r = 1 above, we obtain a contradiction with the minimality of dim η_{1}.

Let φ be a neighbour of a Pfister form τ and let η be the complementary form of φ. By §20 we have φ ⊗ k(φ) ∼ (−η) ⊗ k(φ). We further assume that dim η ≥ dim QL(η) + 2. {In the case char k ≠ 2 this implies dim η ≥ 2.} Now we can form the function field k(η). We have k(φ) ∼_{k} k(τ) and τ ⊗ k(η) ∼ 0. Hence there exists a place from k(φ) to k(η) over k, and so we also have φ ⊗ k(η) ∼ (−η) ⊗ k(η). Conversely we now ask:

Let φ and η be anisotropic forms over k with dim φ > dim η ≥ 2+dim QL(η). Assume that φ⊗k(φ) ∼ η ⊗k(φ) and φ⊗k(η) ∼ η ⊗k(η). Is then φ a Pfister neighbour with complementary form −η?

If char k ≠ 2, this is true, cf. [K_{6}, Th. 8.9]. The proof of this fact again utilizes the argument described in (B), which breaks down in characteristic 2.

Addendum (2009) Problem (B) is now solved. Hoffmann and Laghribi have established Theorem 1 above also in characteristic two, without any restrictions [HL], [EKM, §26].

176 Chapter III. Some Applications

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§25 Leading Form and Degree Function

Let k be a field of arbitrary characteristic and let φ be a form over k, which is just as before understood to be a nondegenerate quadratic form.

In case the characteristic of k is different from 2, we alternatively interpret φ as a bilinear form. In fact we identify φ with the bilinear form β on k^{n} (n = dim φ) which satisfies φ(x) = β(x, x). \{Thus β = \frac{1}{2}B_{φ}, where B_{φ} is the bilinear form associated to φ.} In particular we make the identification

⟨a_{1}, …, a_{n}⟩ = [a_{1}, …, a_{n}] (a_{i} ∈ k^{∗}).

One should bear in mind that with respect to this identification quadratic Pfister forms over k correspond to bilinear Pfister forms over k.

In the rest of this section we will always assume that φ is strictly regular.

Theorem 1. Assume that φ is not hyperbolic. Then the minimum of the dimensions of the kernel forms ker(φ ⊗ L), where L runs through all field extensions of k with φ ⊗ L ≁ 0, is a 2-power 2^{d}.

Proof. If dim φ is odd (and thus char K ≠ 2), this minimum is clearly 1, and the statement holds with d = 0.

Thus assume that dim φ is even. Let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower of φ. Since φ is not hyperbolic we have h ≥ 1.

The kernel form of φ ⊗ K_{h−1} has height 1. By §20, Theorem 5 it is thus of the form aτ with a ∈ K_{h−1}^{∗} and τ a d-fold Pfister form over K_{h−1} for some d ≥ 1. By our generic splitting theory (cf. the end of §9 or §10, Theorem 4) it is clear that 2^{d} is the required minimum.

Definition 1. We call the number d, appearing in Theorem 1, the degree of the form φ and denote it by deg φ. If φ is hyperbolic, we set deg φ = ∞.

According to the proof of Theorem 1, we have deg φ = 0 in case dim φ is odd (and so char k ≠ 2), and deg φ ≥ 1 otherwise. An anisotropic n-fold Pfister form clearly has degree n.

The notion of degree of a form is closely connected to the notion of a “leading form”, which we introduce next.

Definition 2. Let φ be nondegenerate and let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower of φ.

(a) We call any field extension of k, which is specialization equivalent to K_{h−1} over k, a leading field for φ.

(b) Let F be a leading field for φ. By §23 the kernel form ψ of φ⊗F is of the form aτ for dim φ even, and of the form aτ′ with a ∈ F^{∗} and τ a Pfister form for dim φ odd. {Recall: τ′ denotes the pure part of τ.} We call this Pfister form τ over F the leading form of φ over F.

From the proof of Theorem 1 it is clear that in even dimension the degree of φ corresponds to the degree of any leading form τ of φ. In odd dimension there is a connection between the degree of τ and the degree of the evendimensional form φ ⊥ −d(φ), where d(φ) denotes the signed determinant of φ, interpreted as a one-dimensional quadratic form, cf. §2.

Theorem 2. Assume that φ is not split and that dim φ is odd (and so char k ≠ 2 since φ is strictly regular). Let τ be a leading form of φ. Then the forms τ and φ ⊥ −d(φ) have the same degree.

Proof. The form ψ: = φ ⊥ −d(φ) is not split. It has signed determinant^{39} d(ψ) = 1. Let F be a leading field for φ and E a leading field for ψ. We have

φ ⊗ F ∼ aτ′, ψ ⊗ E ∼ bσ,

where a and b are scalars, τ is the leading form of φ and σ is the leading form of ψ. Let n: = deg τ, m: = deg σ = deg ψ. Since d(ψ) = 1 we have d(σ) = 1, and so m ≥ 2. Now d(τ′) = ⟨−1⟩, so that d(φ)⊗F = d(φ⊗F) = ⟨−a⟩. Hence

ψ ⊗ F ∼ aτ′ ⊥ ⟨a⟩ = aτ.

Therefore the kernel form of ψ ⊗ F has dimension 2^{n} and it follows that m ≤ n. On the other hand we have

φ ⊗ E ∼ bσ ⊥ (d(φ) ⊗ E).

The form on the right hand side is of dimension 2^{m}+1. It cannot be split since σ is anisotropic and has dimension ≥ 4. From the generic splitting theory (see for instance §4, Scholium 1) it follows that 2^{m} + 1 ≥ dim(aτ′) = 2^{n} − 1, and so 2^{m} + 2 ≥ 2^{n}. Since m ≤ n and m ≥ 2, we must have m = n.

We now describe a situation where the leading form of φ can be determined.

Theorem 3 (cf. [K_{5}, Th.6.3] for char k ≠ 2). Let n ∈ ℕ. Let φ be an orthogonal sum aτ ⊥ ψ with a ∈ k^{∗}, τ a Pfister form over k of degree n ≥ 1, and ψ a form over k with deg ψ ≥ n + 1. Let F be a leading field for φ.

(i) The leading form of φ over F is τ ⊗ F. (ii) If deg ψ ≥ n + 2, then φ ⊗ F has kernel form (aτ) ⊗ F.

39We write d(ψ) = 1 instead of d(ψ) = ⟨1⟩.

Proof (as in [K_{5}, p.88f]). (i) We may assume that ψ is not split. We will show that φ has degree n.

Let (L_{i} | 0 ≤ i ≤ e) be a generic splitting tower of ψ. Assume that τ ⊗L_{e} ∼ 0. Let s be maximal in^{40} [0, e−1] with τ ⊗L_{s} ≁ 0, so that τ ⊗L_{s} is anisotropic. Let ψ_{s} be the kernel form of ψ ⊗ L_{s}. Then τ ⊗ L_{s}(ψ_{s}) ∼ 0. It then follows from the Norm Theorem and the Subform Theorem in the usual manner that bψ_{s} < τ ⊗ L_{s} for some b ∈ L_{s}^{∗}. Thus, deg ψ = deg ψ_{s} ≤ n, which contradicts our assumption about ψ. Therefore τ ⊗ L_{e} is anisotropic. Hence, φ ⊗ L_{e} has kernel form (aτ) ⊗ L_{e}, and it follows that deg φ ≤ n.

Assume that φ has degree m < n. Let (K_{j} | 0 ≤ j ≤ h) be a generic splitting tower of φ. Now φ ⊗ K_{h−1} is of the form bρ with b ∈ K_{h−1}^{∗} and ρ a Pfister form of degree m over K_{h−1}. We have

ψ ⊗ K_{h−1} ∼ bρ ⊥ (−a)(τ ⊗ K_{h−1}).

The form on the right has dimension 2^{m} + 2^{n} < 2^{n+1}. From deg ψ ≥ n + 1 it follows that ψ ⊗ K_{h−1} ∼ 0.

We obtain bρ ∼ a(τ ⊗ K_{h−1}). Since dim ρ < dim τ, it follows that τ ⊗ K_{h−1} ∼ 0, and so ρ ∼ 0, a contradiction. Thus φ has degree n.

(ii) We have ψ ⊗ K_{h} ∼ (−aτ) ⊗ K_{h} and deg ψ > n. Thus ψ ⊗ K_{h} ∼ 0 and so τ ⊗ K_{h} ∼ 0. Let s be maximal in [0, h − 1] with τ ⊗ K_{s} ≁ 0, so that τ ⊗ K_{s} is anisotropic. From the Norm Theorem and the Subform Theorem it follows again that bφ_{s} < τ ⊗ K_{s} for some b ∈ K_{s}^{∗}. Since φ has degree n we must have s = h − 1 and

φ_{h−1} ≅ b(τ ⊗ K_{h−1}).

Thus τ ⊗ K_{h−1} is the leading form of φ over K_{h−1}. Furthermore,

ψ ⊗ K_{h−1} ∼ [φ ⊥ (−aτ)] ⊗ K_{h−1} ∼ ⟨b, −a⟩ ⊗ τ ⊗ K_{h−1}.

If deg ψ ≥ n + 2, then ψ ⊗ K_{h−1} ∼ 0 and so

φ_{h−1} ≅ (aτ) ⊗ K_{h−1}.

It is already clear from Definition 1 that every form which is Witt equivalent to φ is of the same degree as φ. Therefore we have a degree function

deg: Wq(k) ⟶ ℕ_{0} ∪ {∞}

on the Witt group Wq(k) of strictly regular forms over k, given by deg({φ}): = deg φ. In char k = 2 this function has values in ℕ ∪ {∞}.

The degree function motivates a further definition.

Definition 3. For every n ∈ ℕ we denote by J_{n}(k) the set of all Witt classes ξ of strictly regular forms with deg(ξ) ≥ n.

40For n ∈ ℕ we denote the set of numbers 0, 1, 2, …, n by [0, n].

What does this notation mean for n = 1? Clearly J_{1}(k) is the set of Witt classes of strictly regular forms of even dimension. If char k = 2, we thus have J_{1}(k) = Wq(k). If char k ≠ 2, then Wq(k) = W (k) is a ring and J_{1}(k) is an ideal of W (k), namely the kernel of the dimension index

ν: W (k) ⟶ ℤ/2, {φ} ⟼ dim φ + 2ℤ,

which already made an appearance in §2. This homomorphism ν is also available in characteristic 2. Its kernel is denoted by I(k) and is called the fundamental ideal of W (k). Thus W (k)/I(k) = ℤ/2. In char k ≠ 2 we have I(k) = J_{1}(k).

In more generality we have:

Theorem 4. For every n ∈ ℕ, J_{n}(k) is a W (k)-submodule of Wq(k).

Proof. (As in [K_{5}, p.89f].)

(a) We want to show that J_{n}(k) is closed under addition in Wq(k). We will prove the following equivalent statement: for any two strictly regular forms φ_{1}, φ_{2} over k we have

(∗)  deg(φ_{1} ⊥ φ_{2}) ≥ min(deg φ_{1}, deg φ_{2}).

This statement is trivial when φ_{1} or φ_{2} has odd dimension, and also when φ_{1} ⊥ φ_{2} ∼ 0. In the following we exclude these cases.

Let n: = deg(φ_{1} ⊥ φ_{2}). There exists a field extension L of k such that

ker(φ_{1} ⊗ L ⊥ φ_{2} ⊗ L) = aρ,

for some a ∈ L^{∗} and a Pfister form ρ of degree n. By definition of the degree function we have

deg(φ_{i} ⊗ L) ≥ deg φ_{i} (i = 1, 2).

Therefore it suffices to prove the statement for the forms φ̃_{i}: = φ_{i} ⊗L instead of φ_{i}. If deg φ̃_{2} > n, then the Witt equivalence

φ̃_{1} ∼ aρ ⊥ (−φ̃_{2})

and Theorem 3 give us that φ̃_{1} has degree n. Thus we always have

min(deg φ̃_{1}, deg φ̃_{2}) ≤ n.

(b) W (k) is additively generated by the one-dimensional bilinear forms ⟨a⟩. Since for every strictly regular form φ, the form aφ is of the same degree as φ, it is clear that every J_{n}(k) is stable under multiplication with elements from W (k).

An immediate consequence of Theorem 4 is

Corollary 1. Let φ and ψ be strictly regular forms over k with deg φ ≠ deg ψ. Then

deg(φ ⊥ ψ) = min(deg φ, deg ψ).

It is commonplace to denote the n-th power of the ideal I(k) by I^{n}(k) (instead of I(k)^{n}). It is also customary to leave out the curly brackets {, }, which indicate that not the form itself, but rather its Witt class, is considered. For example, one writes γ ∈ I^{n}(k) to indicate that the Witt class of a nondegenerate bilinear form γ is in I^{n}(k). We will follow this custom.

The ideal I(k) is additively generated by the bilinear forms a⟨1, b⟩ where a, b ∈ k^{∗}. Thus I^{n}(k) is additively generated by the forms aτ where a ∈ k^{∗} and τ is an n-fold bilinear Pfister form which we may in addition assume to be anisotropic. This is also true for n = 0, if we formally write I^{0}(k) = W (k).

The W (k)-module J_{1}(k) is additively generated by the binary forms a [{1 \atop 1} {1 \atop b}]. Thus the W (k)-module I^{n}(k)J_{1}(k) is additively generated by the forms aτ, where a ∈ k^{∗} and τ is a quadratic anisotropic Pfister form of degree n + 1. Thus, by Theorem 4, every element of I^{n}(k)J_{1}(k) is of degree ≥ n + 1. Hence we obtain

Corollary 2. For every n ∈ ℕ_{0} we have

I^{n}(k)J_{1}(k) ⊂ J_{n+1}(k).

Arason and Pfister already showed in 1970, without a generic splitting theory (which did not exist yet at the time), that in characteristic ≠ 2 every anisotropic form φ ≠ 0 which lies in I^{n}(k) is of dimension at least 2^{n} (the so-called Arason-Pfister Hauptsatz [AP]). The proof of Arason-Pfister can be modified in such a way that it also works in characteristic 2, but we will not elaborate that here. {One should of course replace I^{n}(k) by I^{n−1}(k)J_{1}(k).} The statement of our Corollary 2 is equivalent to the Arason-Pfistert Hauptsatz, in every characteristic.

Next we wish to characterize the strictly regular even-dimensional forms φ of degree 1 by employing the discriminant algebra Δ(φ) (cf. the beginning of §13). We begin with some comments on this invariant of φ.

If L/k is a quadratic separable field extension, there exists a generator ω of L/k with minimal equation ω^{2} + ω = a, where 1 + 4a ∈ k^{∗}. The norm form N_{L/k}(x) of L/k has the value matrix [{1 \atop 1} {1 \atop a}] with respect to 1, ω. If char k ≠ 2 then 1, 1 + 2ω is an orthogonal basis with respect to the norm form, which is thus of the form ⟨1, −1 − 4a⟩. Also, (1 + 2ω)^{2} = 1 + 4a.

All of this remains basically true for a = 0 when L is equal to the split quadratic separable algebra k × k. In this case the norm form is hyperbolic instead of anisotropic as before.

This can all be verified easily and so we obtain straight away the following elementary lemma.

Lemma 1. The isomorphism classes of quadratic separable k-algebras L are in one-one correspondence with the isometry classes of binary (= 2dimensional ) Pfister forms τ via L = Δ(τ), τ = N_{L/k}.

We recall further that in characteristic 2 the discriminant algebra Δ(φ) is just the Arf invariant in disguise (see the beginning of §13), while in characteristic ≠ 2 it corresponds to the signed determinant d(φ) since then ⟨1, −d(φ)⟩ is the binary Pfister form associated to Δ(φ). Thus one verifies immediately:

Lemma 2. Let φ_{1} and φ_{2} be two regular forms of even dimension.

(a) Δ(φ_{1} ⊥ φ_{2}) = 1 if and only if Δ(φ_{1}) = Δ(φ_{2}).

(b) Δ(φ_{1} ⊥ φ_{2}) = Δ(φ_{1}) if and only if Δ(φ_{2}) = 1.

Remark. These statements can be placed on a more conceptual footing by endowing the set of isomorphism classes of quadratic separable k-algebras with a multiplication, which turns it into an abelian group QS(k) of exponent 2, and observing that for regular forms φ_{1}, φ_{2} of even dimension, Δ(φ_{1} ⊥ φ_{2}) = Δ(φ_{1}) · Δ(φ_{2}), cf. [B_{1}, Chap.IV], [B_{2}], [Kn_{3}, §10].

Theorem 5. Let φ be a strictly regular form of even dimension. Let τ be the binary Pfister form over k with Δ(φ) = Δ(τ) (cf. Lemma 1).

(a) If Δ(φ) = 1, then deg φ ≥ 2.

(b) If Δ(φ) ≠ 1 (thus τ is anisotropic), then deg(φ) = 1 and φ ∼ τ ⊥ ψ for some strictly regular form ψ of degree ≥ 2. Furthermore, if F is a leading field for φ, then τ ⊗ F is the associated leading form of φ.

Proof. Let F be a leading field for φ, and ρ the associated leading form of φ. Thus ker(φ ⊗ F) = aρ for some scalar a ∈ F^{∗}. If Δ(φ) = 1, then Δ(aρ) = 1. It follows that ρ, and thus also φ, is of degree ≥ 2.

Assume now that Δ(φ) ≠ 1. By Lemma 2 the form ψ: = φ ⊥ τ has discriminant Δ(ψ) = 1. Thus deg ψ ≥ 2, by the first part of the proof. From φ ∼ −τ ⊥ ψ and Theorem 3 it follows that φ has degree 1 and leading form τ ⊗ F.

We note that only part (a) of the Lemma was used. Part (b) will be needed for the first time only in §27.

It follows immediately from this Theorem that J_{2}(k) is the set of all Witt classes {φ} of regular forms φ with dim φ even and Δ(φ) = 1. By the remark after Lemma 2 it follows further that the group QS(k) is isomorphic to the additive group J_{1}(k)/J_{2}(k). We will not need this in the sequel, but it is nevertheless good to know.

We return to an examination of the modules J_{n}(k) for arbitrary n.

Theorem 6. Let φ be a strictly regular quadratic form and α a nondegenerate bilinear form. Assume that α has odd dimension. Then

deg(α ⊗ φ) = deg φ.

Proof. This is clear when the dimension of φ is odd, and also when φ ∼ 0. Therefore we will exclude those cases. Thus let h(φ) > 0 and n: = deg φ ≥ 1. By Theorem 4 we have deg(α ⊗ φ) ≥ n.

We choose a leading field F for φ and obtain φ ⊗ F ∼ aτ for some a ∈ F^{∗} and some n-fold anisotropic Pfister form τ. We also consider some decomposition

α ≅ ⟨b⟩ ⊥ b_{1}ρ_{1} ⊥ … ⊥ b_{r}ρ_{r}

where b, b_{1}, …, b_{r} ∈ k^{∗} and ρ_{1}, …, ρ_{r} are binary bilinear Pfister forms. Hence,

(α ⊗ φ) ⊗ F ∼ bτ ⊥ b_{1}(ρ_{1} ⊗ τ) ⊥ … ⊥ b_{r}(ρ_{r} ⊗ τ),

where we shortened (ρ_{i} ⊗F)⊗τ to ρ_{i} ⊗τ. The ρ_{i} ⊗τ are (n+1)-fold quadratic Pfister forms. By Theorem 4, b_{1}(ρ_{1} ⊗τ) ⊥ … ⊥ b_{r}(ρ_{r} ⊗τ) has degree ≥ n+1. By Theorem 3 (or Corollary 1), (α ⊗ φ) ⊗ F has degree n. It follows that deg(α ⊗ φ) ≤ n and so deg(α ⊗ φ) = n.

We record a much weaker, but still interesting, version of this theorem:

Corollary. Let α be a bilinear form of odd dimension and φ a quadratic strictly regular form with φ ≁ 0. Then α ⊗ φ ≁ 0.

Theorem 7. For arbitrary m ≥ 1, n ≥ 1 we have

I^{m}(k)J_{n}(k) ⊂ J_{m+n}(k).

Proof. (cf. [K_{5}, p.91]). Of course it suffices to show that I(k)J_{n}(k) ⊂ J_{n+1}(k). Since I(k) is generated by the forms ⟨1, a⟩ as an ideal, the theorem boils down to the following statement: If φ is a strictly regular quadratic form of even dimension over k which is not hyperbolic, then, for given a ∈ k^{∗}, the degree of ⟨1, a⟩ ⊗ φ is larger than the degree of φ.

We prove this statement by induction on h(φ). The case h(φ) = 1 is clear. Let h(φ) > 1. If the form α: = ⟨1, a⟩ ⊗ φ splits, the statement is true. Assume now that α ≁ 0. We choose a leading field F for α. Let ρ be the associated leading form of α over F. Since deg(φ ⊗ F) ≥ deg φ it suffices to show that deg ρ > deg(φ ⊗ F). We have h(φ ⊗ F) ≤ h(φ). If we replace k by F and φ by φ ⊗ F, we are reduced to the case

(1)  ⟨1, −a⟩ ⊗ φ ∼ ρ

for some anisotropic Pfister form ρ. Now we must show that deg ρ > deg φ. Assume next that ρ ⊗ k(φ) is anisotropic. We have

⟨1, −a⟩ ⊗ (φ ⊗ k(φ)) ∼ ρ ⊗ k(φ)

and deg(φ ⊗ k(φ)) = deg φ, deg(ρ ⊗ k(φ)) = deg(ρ), h(φ ⊗ k(φ)) = h(φ) − 1. Thus deg ρ > deg φ by the induction hypothesis.

The remaining case is when ρ ⊗ k(φ) splits. By the Norm Theorem and the Subform Theorem we have

(2)  ρ ≅ bφ ⊥ ζ,

for some b ∈ k^{∗} and some form ζ over k. Assume that ζ = 0. Then

⟨1, −a, −b⟩ ⊗ φ ∼ 0

by (1) and (2), contradicting the corollary to the previous theorem. Hence ζ ≠ 0, and so dim φ < dim ρ. Since ρ is a Pfister form, it follows that deg φ < deg ρ.

Nowadays it is known that for every natural number n,

J_{n}(k) = I^{n}(k)

when char k ≠ 2 [OVV], and

J_{n}(k) = I^{n−1}(k)Wq(k)

when char k = 2 [AB]. Thus, in Theorem 7 we have, de facto, I^{m}(k)J_{n}(k) = J_{n+m}(k).

The proofs of these statements are very dissimilar, and go far beyond the scope of this book. The case characteristic ≠ 2 seems to be the more difficult one.

§26 The Companion Form of an Odd-dimensional Regular Form

In this section k is an arbitrary field and φ is always a regular form of odd dimension over k.

If char k ≠ 2, then φ is strictly regular. In characteristic 2 however, φ has a one-dimensional quasilinear part, QL(φ) = [a]. In characteristic ≠ 2 we also interpret φ as a bilinear form in the manner indicated at the beginning of §25.

Theorem 1. Up to isometry there is a unique form ψ over k with the following four properties:

(1) ψ is strictly regular.

(2) Δ(ψ) = 1.

(3) φ < ψ.

(4) dim ψ = dim φ + 1.

Proof. If char k ≠ 2, we need ψ = φ ⊥ ⟨b⟩ for some b ∈ k^{∗} such that the condition d(ψ) = 1 is satisfied. So we take b such that ⟨b⟩ = −d(φ).

Now let char k = 2. We choose an orthogonal decomposition φ: = χ ⊥ [a]. Thus χ is strictly regular and QL(φ) = [a]. We need ψ = χ ⊥ α for some binary strictly regular form α, to be determined. The condition χ ⊥ [a] < ψ is equivalent with [a] < α (cf. §17, Theorem 3). Thus, by §25, Lemma 2(a), α should be chosen in such a way that a ∈ D(α) and Δ(α) = Δ(χ). By §25, Lemma 1, there is exactly one binary Pfister form τ with Δ(τ) = Δ(χ). We must take α = aτ.

Definition 1. We call the form ψ, described in Theorem 1, the companion of φ. We denote this form, which is uniquely determined by φ, by Com(φ).

Remark. The mapping φ ↦ Com(φ) satisfies the following rules:

(1) If α is a strictly regular form with Δ(α) = 1, then Com(φ ⊥ α) = Com(φ) ⊥ α. In particular, Com(γ) ∼ Com(φ) for every regular form γ ∼ φ.

(2) Com(aφ) = aCom(φ) for every a ∈ k^{∗}.

(3) Com(φ ⊗ L) = Com(φ) ⊗ L for every field extension L ⊃ k.

(4) If char k ≠ 2, then Com(φ) = φ ⊥ −d(φ).

Proof. The rules (1) – (3) follow immediately from the definition of a companion. Rule (4) is clear from the proof of Theorem 1 above.

Conversely, every strictly regular form ψ of even dimension and discriminant 1 generally acts as the companion of many regular forms φ. More precisely the following theorem holds, which can easily be verified upon consulting the explanations about polars in §16.

Theorem 2.

(i) Let ψ be a strictly regular form of even dimension over k with Δ(ψ) = 1. For every a ∈ D(ψ) the polar φ: = Pol_{ψ}([a]) (cf. §16) is a strictly regular form of odd dimension with Com(φ) = ψ. If k is of characteristic 2, then [a] = QL(φ). If k is of characteristic ≠ 2, then ⟨a⟩ = −d(φ).

(ii) The isometry classes (φ) of strictly regular forms φ of odd dimension over k are in one-one correspondence with the pairs ((ψ), [a]), where (ψ) is the isometry class of a strictly regular form of even dimension and discriminant 1 (thus {ψ} ∈ J_{2}(k)) and a ∈ D(ψ), via φ = Pol_{ψ}[a] and ψ = Com(φ), [a] = QL(φ) for char k = 2, ⟨a⟩ = −d(φ) for char k ≠ 2.

The following theorem gives another method for retrieving φ from ψ and d(φ) from QL(φ), respectively, by means of the Cancellation Theorem (§6, Theorem 2 – for fields).

Theorem 3. Let ψ = Com(φ).

(a) If char k ≠ 2, then φ ⊥ H ≅ ψ ⊥ d(φ).

(b) If char k = 2, then φ ⊥ H ≅ ψ ⊥ QL(φ).

Proof. (a) This is clear since ψ = φ ⊥ −d(φ) in this case.

(b) Since QL(φ) = [a] we have

φ ⊥ H ≅ Pol_{ψ}([a]) ⊥ H ≅ Pol_{ψ⊥H}([a]).

Furthermore, ψ ⊥ H ≅ ψ ⊥ [{a \atop 1} {1 \atop 0}]. Thus,

Pol_{ψ⊥H}([a]) ≅ ψ ⊥ [a].

Next we present necessary and sufficient conditions for the companion Com(φ) to be isotropic or split.

Theorem 4. The following statements are equivalent:

(i) Com(φ) is isotropic. (ii) There exists an element b ∈ k^{∗} and a strictly regular form χ of even

dimension with Δ(χ) = 1 and φ ≅ χ ⊥ [b]. (iii) There exists a strictly regular form χ of even discriminant with Δ(χ) =

1, χ < φ and 1 + dim χ = dim φ.

§26 The Companion Form of an Odd-dimensional Regular Form 187

If (i) – (iii) hold, then Com(φ) = χ ⊥ H and, furthermore, [b] = QL(φ) when char k = 2 and ⟨b⟩ = d(φ) when char k ≠ 2.

Proof. If χ is a strictly regular form with χ < φ, dim χ = dim φ − 1, then we have a decomposition φ ≅ χ ⊥ [b] for some b ∈ k^{∗}. This establishes the equivalence (ii) ⇔ (iii). We also see that then QL(φ) = [b] in characteristic 2 and d(φ) = ⟨b⟩ in characteristic ≠ 2.

Let ψ: = Com(φ). By Theorem 3 we have an isometry φ ⊥ H ≅ ψ ⊥ [b]. If ψ is isotropic, we have in addition a decomposition ψ ≅ χ ⊥ H and we have Δ(χ) = 1. It follows that φ ≅ χ ⊥ [b]. Conversely, if φ ≅ χ ⊥ [b] with Δ(χ) = 1, then the form χ ⊥ H ≅ χ ⊥ [{b \atop 1} {1 \atop 0}] satisfies the conditions of Theorem 1, and so χ ⊥ H ≅ ψ.

Theorem 5. Com(φ) splits if and only if φ splits.

Proof. If φ ≅ r × H ⊥ [a], then the form ψ: = r × H ⊥ [{a \atop 1} {1 \atop 0}] ≅ (r + 1) × H satisfies the four conditions of Theorem 1. Hence, ψ = Com(φ). Conversely, if Com(φ) is hyperbolic, then φ ⊥ H splits by Theorem 3. This implies that φ splits.

As a digression we remark that the unicity statement contained in Theorem 4 leads to a new kind of cancellation theorem:

Theorem 6. Let a ∈ k and let χ_{1}, χ_{2} be two strictly regular forms of even dimension with Δ(χ_{1}) = Δ(χ_{2}) and χ_{1} ⊥ [a] ≅ χ_{2} ⊥ [a]. Then χ_{1} ≅ χ_{2}.

Proof. This is trivial when a = 0. Thus let a ≠ 0. If Δ(χ_{1}) = Δ(χ_{2}) = 1, the statement follows immediately from Theorem 4. If only Δ(χ_{1}) = Δ(χ_{2}), then Δ(χ_{1} ⊥ χ_{2}) = Δ(χ_{2} ⊥ χ_{2}) = 1 by §25, Lemma 2. We have χ_{1} ⊥ χ_{2} ⊥ [a] ≅ χ_{2} ⊥ χ_{2} ⊥ [a], and deduce that χ_{1} ⊥ χ_{2} ≅ χ_{2} ⊥ χ_{2}, and finally that χ_{1} ≅ χ_{2} with the old Cancellation Theorem §6, Theorem 2.

We make a general statement about the Witt indices of φ and Com(φ):

Theorem 7. We have

ind(φ) ≤ ind(Com(φ)) ≤ 1 + ind(φ).

Proof. Let ψ: = Com(φ). Since φ < ψ we have ind(φ) ≤ ind(ψ). On the other hand we have an isometry φ ⊥ H ≅ ψ ⊥ [a] by Theorem 3. Hence

ind(ψ) ≤ ind(ψ ⊥ [a]) = 1 + ind(φ).

Theorem 8. h(φ) ≤ 2h(Com(φ)).

Proof. Let ψ: = Com(φ), h: = h(φ), e: = h(ψ). Then h + 1 is the number of possible Witt indices of φ ⊗ K, where K traverses all field extensions of k, thus h + 1 = card SP(φ). Similarly, e + 1 = card SP(ψ). Furthermore we have Com(φ ⊗ K) = ψ ⊗ K. If ind(ψ ⊗ K) = j, then ind(φ ⊗ K) = j or j + 1 by Theorem 7. However, if ind(ψ ⊗ K) has the highest possible value \frac{1}{2} dim ψ, then ind(φ⊗K) also has the highest possible value \frac{1}{2} dim ψ−1 by Theorem 5. It follows that h + 1 ≤ 2e + 1, and so h ≤ 2e.

In §25 we associated to every strictly regular form, which is not split, a leading form. We will now also introduce a leading form for regular forms of odd dimension, in characteristic 2. First some preliminaries.

Definition 2. Let τ be a (quadratic) Pfister form over k. Since 1 ∈ D(τ) we can form the polar Pol_{τ} [1], which we call the unit polar of τ and denote by τ′.

Remarks.

(1) If dim τ = 2 and char k = 2, then τ′ = [1], and forming the unit polar is not interesting. On the other hand, if dim τ ≥ 4, then clearly τ is the strictly regular companion of τ′ and τ′ is the regular form of odd dimension associated to the pair (τ, [1]) in the sense of Theorem 2.

(2) For an arbitrary Pfister form σ over k we introduced the pure part of σ, denoted by σ′, earlier (§16). In char k ≠ 2 this pure part is identical to the unit polar.

It stands to reason to look for a relationship between the generic spitting behaviour of the form φ and its companion Com(φ) = ψ, and in particular to compare the leading forms of φ and ψ (cf. §23, Def. 2). A result in this vein, presented below, can already be found in [K_{5}, §5] for the characteristic ≠ 2 case. A partial result in characteristic ≠ 2 was anticipated in §25, Theorem 2.

Theorem 9. Let φ be a regular form of odd dimension over k and ψ: = Com(φ) its companion. Let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower of φ and (L_{s} | 0 ≤ s ≤ e) a generic splitting tower of ψ. We assume that φ, and so (by Theorem 5) also ψ, does not split. Let τ be the leading form of φ over K_{h−1} and σ the leading form of ψ over L_{e−1}. Then we claim:

(i) K_{h} ∼_{k} L_{e}. (ii) There is a place λ: L_{e−1} → K_{h−1} ∪ ∞ over k. (iii) σ has good reduction with respect to every such place λ, and λ_{∗}(σ) ≅ τ.

In particular, deg τ = deg σ = deg ψ.

Proof. For every field extension L of k we have Com(φ ⊗ L) = ψ ⊗ L. Thus ψ ⊗ L splits if and only if φ ⊗ L splits by Theorem 5. Hence K_{h} and L_{e} are specialization equivalent over k.

The kernel form φ_{h−1} of φ ⊗ K_{h−1} is of the form aτ′ for some a ∈ K_{h−1}^{∗}. {Incidentally, a can be chosen in k, but we do not need that now.} We have

(∗)  ψ ⊗ K_{h−1} = Com(φ ⊗ K_{h−1}) ∼ Com(φ_{h−1}) = aτ.

It follows that deg τ ≥ deg ψ. On the other hand, the kernel form ψ_{e−1} of ψ ⊗ L_{e−1} is of the form bσ for some b ∈ L_{e−1}^{∗}. Hence, φ ⊗ L_{e−1} ∼ bσ ⊥ [a], where [a] = QL(φ) in characteristic 2 and [a] = d(φ) in characteristic ≠ 2. Since the form σ is anisotropic and of dimension at least 4, bσ ⊥ [a] cannot split.

It follows from the general generic splitting theory that dim σ+1 ≥ dim τ′, hence dim σ + 2 ≥ dim τ. Earlier we established that dim σ ≤ dim τ. Since dim σ ≥ 4 and the integers dim σ and dim τ are both 2-powers, it follows that dim σ = dim τ. Now (∗) tells us, by the generic splitting theory, that there exists a place λ from L_{e−1} to K_{h−1} over k, and furthermore that bσ has good reduction with respect to every such place λ and that λ_{∗}(bσ) = aτ. Since bσ has good reduction with respect to λ, bσ will certainly represent a unit c in the valuation ring 𝔬_{λ}. Then we have bσ ≅ cσ, and it follows that σ has good reduction with respect to λ, and that λ_{∗}(σ) = λ(c)aτ. This forces λ_{∗}(σ) = τ.

As an illustration of the results obtained so far, we classify the anisotropic regular forms of odd dimension whose strictly regular companion is of height 1.

Theorem 10. Let φ be anisotropic. The companion ψ: = Com(φ) is of height 1 if and only if one of the following two cases happens: (A) φ ≅ aτ′ for some anisotropic Pfister form τ of degree ≥ 2 and some

a ∈ k^{∗}. (B) φ ≅ aτ ⊥ [b] for some anisotropic Pfister form τ of degree ≥ 2, some

a ∈ k^{∗} and b ∈ k^{∗} with −ab ∉ D(τ).

In case (A) we have ψ ≅ aτ, and so ψ is anisotropic. In case (B) we have ψ ≅ aτ ⊥ H, and so ψ is isotropic. In case (A) φ is of height 1, in case (B) of height 2. In both cases φ is excellent.

Proof. (1) Let τ be a Pfister form of degree ≥ 2 and let a and b be elements of k^{∗}. If φ ≅ aτ′, then ψ = aτ, and so h(ψ) = 1. Furthermore we have h(φ) = 1 in this case. If φ ≅ aτ ⊥ [b] and −ab ∉ D(τ), then φ is anisotropic, and ψ ≅ aτ ⊥ [{b \atop 1} {1 \atop 0}] ≅ aτ ⊥ H. Therefore ψ is also of height 1 in this case. Furthermore,

φ ⊥ bτ′ ≅ a⟨1, ab⟩ ⊗ τ.

Thus φ is a neighbour of the anisotropic Pfister form ⟨1, ab⟩ ⊗ τ with complementary form bτ′. Hence φ is excellent of height 2.

(2) Assume now that h(ψ) = 1. By Theorem 7 we have ind(ψ) ≤ 1.

Case (A): ψ is anisotropic. Then ψ ≅ cτ for some c ∈ k^{∗} and some Pfister form τ of degree ≥ 2. If char k = 2, and QL(φ) = [a], then we may choose c = a and have φ = Pol_{ψ}([a]) ≅ aτ′. If char k ≠ 2, then φ is a strictly regular subform of ψ, and so we obtain again that φ ≅ aτ′ for some a ∈ k^{∗}.

Case (B): ψ is isotropic. Then ψ ≅ aτ ⊥ H for some a ∈ k^{∗} and some Pfister form τ of degree 2. By Theorem 3 we have φ ⊥ H ≅ ψ ⊥ [b] for some b ∈ k^{∗}. Hence, φ ≅ aτ ⊥ [b]. Since φ is anisotropic it follows that −ab ∉ D(τ).

Next we will more closely consider and compare the generic splitting towers of φ and ψ: = Com(φ). Assume as before that dim φ is odd. Let further ak^{∗2} be the square class with d(φ) = ⟨−a⟩ in case char k ≠ 2, and QL(φ) = [a] in case char k = 2. Thus a ∈ D(ψ) and φ = Pol_{ψ}([a]). We need two lemmata.

Lemma 1. Let φ_{0}: = ker(φ) and ψ_{0}: = ker(ψ). The following statements are equivalent:

(i) Com(φ_{0}) = ψ_{0}. (ii) Com(φ_{0}) is anisotropic. (iii) ind(ψ) = ind(φ). (iv) a ∈ D(ψ_{0}).

Proof. Let r: = ind(φ), hence φ ≅ φ_{0} ⊥ r × H and χ: = Com(φ_{0}). From Remark (1) above it follows that

ψ ≅ χ ⊥ r × H.

This is the Witt decomposition of ψ if and only if ind(ψ) = r or, equivalently, if and only if χ is anisotropic. This already proves the equivalence of (i) – (iii).

Furthermore, d(φ_{0}) = d(φ) = ⟨−a⟩ if char k ≠ 2, and QL(φ_{0}) = QL(φ) = [a] if char k = 2. Thus a ∈ D(χ) and φ_{0} = Pol_{χ}([a]). If χ = ψ_{0}, then certainly a ∈ D(ψ_{0}). This establishes (i) ⇒ (iv).

Finally, assume that a ∈ D(ψ_{0}). Of course ψ_{0} ∼ χ. Hence, Pol_{ψ_{0}}([a]) ∼ Pol_{χ}[a] = φ_{0}, and so φ_{0} ≅ Pol_{ψ_{0}}([a]), since the forms φ_{0} and Pol_{ψ_{0}}([a]) are both anisotropic. Therefore, ψ_{0} = Com(φ_{0}). This shows (iv) ⇒ (i).

We assume from now on that φ, and thus ψ, is not split.

Lemma 2. Let ψ = Com(φ) be anisotropic. Let ψ_{1} denote the kernel form of ψ ⊗ k(ψ).

(i) There exists a place from k(ψ) to k(φ) over k. (ii) k(φ) ∼_{k} k(ψ) if and only if a ∈ D(ψ_{1}) or ind(ψ ⊗ k(ψ)) ≥ 2. (iii) If k(φ) is not specialization equivalent to k(ψ) over k, then φ ⊗ k(ψ) is

anisotropic.

Proof. (i) is clear since φ is a subform of ψ. Let K: = k(ψ). We have Com(φ⊗ K) = Com(ψ) ⊗ K. By the general theory of generic splitting it is clear that k(φ) is specialization equivalent to K over k if and only if φ ⊗ K is isotropic.

A priori we have ind(φ⊗K) = ind(ψ⊗K) or ind(φ⊗K) = ind(ψ⊗K)−1 (by Theorem 7). If ind(ψ ⊗K) ≥ 2, then φ⊗K is definitely isotropic. Assume now that ind(ψ ⊗ K) = 1, thus ψ_{K} ≅ ψ_{1} ⊥ H. The form φ ⊗ K is of index 0 or 1. By Lemma 1, φ⊗K is of index 1 if and only if a ∈ D(ψ_{1}). This happens if and only if φ ⊗ K is isotropic.

Let h: = h(φ) and e: = h(ψ). Further, let (L_{s} | 0 ≤ s ≤ e) be a generic splitting tower of ψ = Com(φ), and let ψ_{s}: = ker(ψ ⊗ L_{s}), η_{s}: = ker(φ ⊗ L_{s}). We know that φ⊗L_{e} splits, so that η_{e} is one-dimensional and thus η_{e} = [−a].

We are looking for an integer m ∈ [0, e], as large as possible, such that (L_{s} | 0 ≤ s ≤ m) is a truncated generic splitting tower of φ, which signifies that there is a generic splitting tower (K_{r} | 0 ≤ r ≤ h) of φ with h ≥ m and K_{r} = L_{r} for 0 ≤ r ≤ m.

Theorem 11. Let a ∈ D(ψ_{m}) for some m ∈ [0, e − 1].

(a) (L_{s} | 0 ≤ s ≤ m) is a truncated generic splitting tower of φ. Furthermore, ind(φ ⊗ L_{s}) = ind(ψ ⊗ L_{s}) and Com(η_{s}) = ψ_{s} for 0 ≤ s ≤ m.

(b) If, in addition, a ∉ D(ψ_{m+1}), then ind(φ ⊗ L_{m+1}) = ind(ψ ⊗ L_{m+1}) − 1 and Com(η_{m+1}) ≅ ψ_{m+1} ⊥ H. The following alternative also holds: if ind(ψ_{m} ⊗ L_{m+1}) ≥ 2, then (L_{s} | 0 ≤ s ≤ m + 1) is a truncated generic splitting tower of φ. On the other hand, if ind(ψ_{m} ⊗ L_{m+1}) = 1, then η_{m} ⊗ L_{m+1} is anisotropic, and so η_{m} ⊗ L_{m+1} = η_{m+1} and

ind(φ ⊗ L_{m+1}) = ind(φ ⊗ L_{m}) = ind(ψ ⊗ L_{m}).

Proof. By induction on m. Lemma 1 and Lemma 2 give us the case m = 0. Thus let m > 0. From the induction hypothesis we know that (L_{s} | 0 ≤ s ≤ m − 1) is a truncated generic splitting tower of φ and that ind(φ ⊗ L_{s}) = ind(ψ ⊗ L_{s}) and Com(η_{s}) = ψ_{s} for 0 ≤ s ≤ m − 1. Let K: = L_{m−1}. By assumption we have L_{m} ∼_{K} K(ψ_{m−1}) and a ∈ D(ψ_{m}). It follows that the element a is also represented by ker(ψ_{m−1} ⊗ K(ψ_{m−1})). Furthermore, ψ_{m−1} = Com(η_{m−1}). From Lemma 2 it follows that K(ψ_{m−1}) ∼_{K} K(η_{m−1}), and so L_{m} ∼_{K} K(η_{m−1}). Hence, (L_{s} | 0 ≤ s ≤ m) is a truncated generic splitting tower of φ. Applying Lemma 1 to η_{m−1} ⊗ L_{m} further shows that Com(η_{m}) = ψ_{m}, and finally that η_{m−1} ⊗ L_{m} and ψ_{m−1} ⊗ L_{m} have the same index. By the induction hypothesis φ ⊗ L_{m−1} and ψ ⊗ L_{m−1} also have the same index. Therefore φ ⊗ L_{m} and ψ ⊗ L_{m} have the same index. This proves part (a) of the theorem. Applying Lemma 2 again, but this time to η_{m} and ψ_{m} = Com(η_{m}), gives part (b).

Theorem 12. We have h ≥ e. {Recall that h: = h(φ), e: = h(ψ). We assume that h > 0, e > 0.} If a ∉ D(ψ_{0}), then we even have h ≥ e + 1. The following statements are equivalent:

(i) a ∈ D(ψ_{e−1}). (ii) (L_{s} | 0 ≤ s ≤ e) is a generic splitting tower of φ. (In particular, h = e.) (iii) Com(η_{s}) = ψ_{s} for 0 ≤ s ≤ e − 1. (iv) ind(φ ⊗ L_{s}) = ind(ψ ⊗ L_{s}) for 0 ≤ s ≤ e − 1.

(v) L_{e−1} is a leading field for φ.

Proof. We distinguish three cases.

Case 1: a ∉ D(ψ_{0}). For every s ∈ [0, e] we establish the following. We have a ∉ D(ψ_{s}), and so ψ_{s} ⊥ [−a] is anisotropic. By Theorem 3 we have ψ ⊥ [−a] ≅ φ ⊥ H. Thus ψ_{s} ⊥ [−a] ∼ η_{s} by the anisotropy of both forms, hence η_{s} ≅ ψ_{s} ⊥ [−a]. For s = e − 1 we obtain η_{e−1} ≅ bσ ⊥ [−a] for some b ∈ L_{e−1}^{∗} and σ the leading form of ψ over L_{e−1}.

This form η_{e−1} is a neighbour of the Pfister form ⟨1, −ab⟩ ⊗ σ (which is thus anisotropic) and has ζ: = −aσ′ as complementary form, and thus (−aσ′)⊗L_{e−1}(η_{e−1}) as first higher kernel form. The kernel form of φ⊗L has a different dimension for each of the fields L = L_{0}, L_{1}, …, L_{e}, L_{e−1}(η_{e−1}). Hence h > e.

Case 2: a ∈ D(ψ_{e−1}). Since ψ_{e} = 0, and thus a ∉ D(ψ_{e}), we may apply Theorem 11. By part (a) of said theorem, (L_{s} | 0 ≤ s ≤ e − 1) is a truncated generic splitting tower of φ and Com(η_{s}) = ψ_{s} for every s ∈ [0, e − 1]. In particular, η_{e−1} is not split. But η_{e−1} ⊗ L_{e} splits, since φ ⊗ L_{e} splits. Thus, by part (b) of Theorem 11, (L_{s} | 0 ≤ s ≤ e) is a truncated generic splitting tower of φ. Since φ ⊗ L_{e} splits, this tower is actually a full generic splitting tower of φ.

Case 3: The rest. Now there exists a largest integer m ∈ [0, e − 2] such that a ∈ D(ψ_{m}). By Theorem 11(a), (L_{s} | 0 ≤ s ≤ m) is a truncated generic splitting tower of φ and Com(η_{s}) = ψ_{s} for 0 ≤ s ≤ m. Furthermore, by Theorem 11(b), Com(η_{m+1}) = ψ_{m+1} ⊥ H, and we have the following alternatives:

Case 3(a): ind(ψ_{m} ⊗ L_{m+1}) ≥ 2. Now (L_{s} | 0 ≤ s ≤ m + 1) is a truncated generic splitting tower of φ.

Case 3(b): ind(ψ_{m} ⊗ L_{m}) = 1. Now η_{m} ⊗ L_{m+1} = η_{m+1}.

(L_{s} | m + 1 ≤ s ≤ e) is a generic splitting tower of Com(η_{m+1}) ∼ ψ_{m+1} ⊥ H, and we have a ∉ D(ψ_{m+1}). For this form ψ_{m+1} we are in Case 1, discussed above, and conclude that h(η_{m+1}) > h(ψ_{m+1}) = e − (m + 1), and so h(η_{m+1}) ≥ e − m. In Case 3(a) we have h(η_{m+1}) = h − (m + 1), and we obtain h ≥ e + 1. In case 3(b) we have η_{m+1} ≅ η_{m} ⊗ L and we may conclude that h(η_{m+1}) ≤ h(η_{m}) = h − m. Therefore, e − m ≤ h − m and so e ≤ h.

Thus e ≤ h in all cases, and in Case 2 we even have e < h. Our analysis shows furthermore that all the conditions (i) – (iv) of the theorem hold in Case 2, and are thus equivalent. The implication (ii) ⇒ (v) is trivial. Assume (v), then η_{e−1} ≅ aτ′ where τ is the leading form of φ over L_{e−1}. Thus ψ_{e−1} ∼ Com(aτ′) = aτ and then ψ_{e−1} ≅ aτ, since the forms ψ_{e−1} and τ are both anisotropic. Hence a ∈ D(ψ_{e−1}). This proves the implication (v) ⇒ (i).

Note that the analysis, carried out above, also establishes the following:

Corollary. If there exists an m ∈ [0, e − 1] with a ∈ D(ψ_{m}), a ∉ D(ψ_{m+1}), and ind(ψ_{m} ⊗ L_{m}) ≥ 2, then h > e.

§27 Definability of the Leading Form over the Base Field

In the following φ denotes a regular (quadratic) form over an arbitrary field k, F denotes a leading field for φ and σ denotes the associated leading form of φ.

Definition 1. We say that the leading form of φ is defined over k if there exists a form τ over k such that σ ≅ τ ⊗ F.

Obviously this property is independent of the choice of the leading field F. Moreover, τ is to a large extent uniquely determined by σ. We namely have the following

Theorem 1 ([K_{6}, Prop. 9.2] for char ≠ 2). Assume that the leading form of φ is defined over k.

(i) If the dimension of φ is even, then there exists up to isometry precisely one form τ over k such that τ ⊗ F ≅ σ.

(ii) If the dimension of φ is odd, then there exists up to isometry precisely

one form τ over k such that 1 ∈ D(τ) and τ ⊗ F ≅ σ. (iii) In both cases τ is a Pfister form.

Proof. We choose a generic splitting tower (K_{r} | 0 ≤ r ≤ h) of φ and assume without loss of generality that F = K_{h−1}. Let τ_{1} and τ_{2} be forms over k such that

τ_{1} ⊗ K_{h−1} ≅ τ_{2} ⊗ K_{h−1} ≅ σ.

Assume that τ_{1} ≅ τ̸_{2}. We must have h ≥ 2. Let s be the largest integer in [0, h − 2] such that τ_{1} ⊗ K_{s} ≅ τ̸_{2} ⊗ K_{s}, and let ζ be the kernel form of [τ_{1} ⊥ (−τ_{2})] ⊗ K_{s}. Then ζ ≠ 0 and the dimension of ζ is even. For 0 ≤ r ≤ h let φ_{r} denote the kernel form of φ ⊗ K_{r}. The form ζ is split by K_{s}(φ_{s}). By the usual argument (the Norm and Subform Theorems) we thus have a decomposition

φ_{s} ⊥ ψ ≅ cζ

for some c ∈ K_{s}^{∗} and some form ψ over K_{s}. It follows that

φ_{s} ⊗ K_{s+1} ∼ (−ψ) ⊗ K_{s+1},

and thus dim ψ ≥ dim φ_{s+1}. We obtain

(∗)  2 dim τ_{1} ≥ dim ζ ≥ dim φ_{s} + dim φ_{s+1} ≥ 2 dim φ_{s+1} + 2.

If dim φ is even, this gives a contradiction and so τ_{1} ≅ τ_{2}.

Assume now that dim φ is odd and moreover that [1] < τ_{1} and [1] < τ_{2}. Then τ_{1} ⊥ (−τ_{2}) is isotropic and we obtain from (∗) that

2 dim τ_{1} ≥ dim ζ + 2 ≥ 2 dim φ_{s+1} + 4,

which is again a contradiction, so that τ_{1} ≅ τ_{2} as before.

In addition there always exists a form τ over k such that 1 ∈ D(τ) and τ ⊗ K_{h−1} ≅ σ (also when dim φ is odd). Namely, if η is a form over k such that η ⊗ K_{h−1} ≅ σ, and if an element c ∈ D(η) is chosen, then cη is such a form τ.

Now let τ be the uniquely determined form over k with 1 ∈ D(τ) and τ ⊗ K_{h−1} ≅ σ. Further, let t = (t_{1}, …, t_{n}) be an n-tuple of unknowns over k with n: = dim τ. We consider the forms φ̃: = φ ⊗ k(t) and τ̃: = τ ⊗ k(t). By §13, Theorem 3, (K_{r}(t) | 0 ≤ r ≤ h) is a generic splitting tower of φ̃ and τ̃ ⊗ K_{h−1}(t) is the associated leading form of φ̃ ⊗ k(t) (also in the event that dim φ is odd). Since this leading form is a Pfister form we have

τ̃ ⊗ K_{h−1}(t) ≅ [τ(t)τ̃] ⊗ K_{h−1}(t).

The form τ(t)τ̃ also represents 1. It follows from what we already proved that τ(t)τ̃ ≅ τ̃. This indicates that τ is strongly multiplicative, and thus is a Pfister form (cf. §20).

Remark. If dim φ is odd, we cannot dispense with the assumption that 1 ∈ D(τ) in assertion (ii), as the following example shows. Let ρ be a Pfister form, a ∈ k^{∗} and ⟨1, a⟩ ⊗ ρ anisotropic. Then the form φ: = ρ ⊥ [a] is a neighbour of ⟨1, a⟩ ⊗ ρ with complementary form

η: = Pol_{ρ⊥aρ}(ρ ⊥ [a]) = Pol_{aρ}([a]) = aPol_{ρ}([1]) = aρ′.

Thus φ has height 2 and leading form ρ ⊗ F, for instance with F = k(φ). However, we do have ρ ⊗ F ≅ (−aρ) ⊗ F, which does not contradict our theorem since 1 ∉ D(−aρ).

Definition 2. Assume that h(φ) > 0 and that the leading form σ of φ is defined over k. Let τ be the Pfister form over k with σ ≅ τ ⊗ F. Then we say that the leading form of φ is defined over k by τ.

Examples.

(1) If φ is excellent, then the kernel form of φ ⊗ F is definable over k by §22, Theorem 3 (for this terminology, see §24(C)). This implies, also when dim φ is odd, that the leading form of φ is defined over k.

(2) Let φ ∼ aτ ⊥ ψ with a ∈ k^{∗}, τ a Pfister form of degree n ≥ 1, and ψ strictly regular with deg ψ ≥ n + 1. By §25, Theorem 3 the leading form of φ is defined over k by τ.

§27 Definability of the Leading Form over the Base Field 195

(3) Let dim φ be odd and Δ(φ) ≠ 1. Let τ be the anisotropic binary Pfister form with Δ(τ) = Δ(φ) (cf. §25, Lemma 1). The form ψ: = φ ⊥ (−τ) has discriminant Δ(ψ) = 1, and thus degree ≥ 2 (cf. §25, Satz 5). We have φ ∼ τ ⊥ ψ and deg τ = 1. Thus, by the previous example, the leading form of φ is defined over k by τ.

In the search for forms whose leading form is defined over the ground field, we may restrict ourselves to forms of even dimension by the following theorem.

Theorem 2 ([K_{6}, Prop. 9.4] for char k ≠ 2). Assume that φ is not split and that dim φ is odd. Let τ be an anisotropic Pfister form over k (of degree ≥ 2). The leading form of φ is defined over k by τ if and only if this is the case for the companion Com(φ).

Proof. Let ψ: = Com(φ). If the leading form of ψ is defined over k by τ, then §26, Theorem 9 shows that the same is true for φ {although the leading fields of φ and ψ might not be equivalent, see Theorem 4 below.}

Assume now that the leading form of φ is defined over k by τ. We must show that the same is true for ψ. Let F be a leading field for φ, and let QL(φ) = [a] for char k = 2, d(φ) = ⟨−a⟩ for char k ≠ 2, where a ∈ k^{∗}. In both cases we have φ ⊥ H ≅ ψ ⊥ [−a] by §26, Theorem 3.

Furthermore, ker(φ ⊗ F) = c(τ′ ⊗ F) for some c ∈ F^{∗}. We see that QL(φ ⊗ F) = [c] for char k = 2, d(φ ⊗ F) = ⟨−c⟩ for char k ≠ 2. Therefore c and a represent the same square class of F, and we may write

ker(φ ⊗ F) = (aτ′) ⊗ F.

Let E be a leading field for ψ and σ the leading form of ψ over E. By §26, Theorem 9, σ and τ have the same dimension ≥ 4. The kernel form of ψ ⊗ E is bσ for some b ∈ E^{∗}.

Suppose for the sake of contradiction that σ is not isomorphic to τ̃: = τ ⊗ E. Then σ ⊗ E(τ̃) is anisotropic. We introduce the form

α: = b(σ ⊗ E(τ̃)) ⊥ [−a]

over E(τ̃) and know from above that φ ⊗ E(τ̃) ∼ α and dim α = dim τ + 1.

And now for a little thought experiment. Let T be a field extension of E(τ̃) for which α ⊗ T is isotropic. Then the kernel form of φ ⊗ T, which coincides with the kernel form of α ⊗ T, has dimension ≤ dim τ′. Thus there exists a place from F to T over k. Since φ ⊗ F ∼ (aτ′) ⊗ F and τ ⊗ T splits, it follows that

α ⊗ T ∼ φ ⊗ T ∼ a(τ′ ⊗ T) ∼ [−a].

By the definition of α and the Cancellation Theorem §26, Theorem 6 we obtain σ ⊗ T ∼ 0.

This thought experiment shows for starters that α itself must be anisotropic since σ ⊗ E(τ̃) is anisotropic. (Take T = E(τ̃).) Secondly, it shows that α splits over every field extension of E(τ̃), over which α is isotropic. Thus α has height 1. By §23 α is up to a scalar factor the unit polar of a Pfister form. Hence dim α + 1 is a 2-power. This is the contradiction we were looking for because dim α + 1 = dim τ + 2 cannot be a 2-power since dim τ is a 2-power ≥ 4. This proves σ ≅ τ ⊗ E.

We come to the main theorem of this section.

Theorem 3 ([K_{6}, Th. 9.6] for char k ≠ 2). Let n ∈ ℕ, let τ be a Pfister form of degree n over k and let φ be a non-split (strictly) regular form of even dimension over k. The following statements are equivalent:

(i) The leading form of φ is defined over k by τ. (ii) φ ≡ τ mod J_{n+1}(k). {Recall: J_{r}(k) is the module of Witt classes of strictly regular forms over k of degree ≥ r.}

Proof. The implication (ii) ⇒ (i) was already established in §25, Theorem 3. Assume now that (i) holds. We want to show that the form ψ: = φ ⊥ (−τ) has degree > n. In order to achieve this we will for the first time make serious use of the theory developed in §13.

We assume without loss of generality that ψ is not hyperbolic. Let (K_{r} | 0 ≤ r ≤ h) be a regular generic splitting tower of φ (cf. §13, Definition 5) and let φ_{r}: = ker(φ ⊗ K_{r}). For every r ∈ [0, h] we consider the field compositum K_{r} ·k(τ) over k in accordance with §13, Definition 6. Let J be the set of all r ∈ [0, h] such that φ_{r} ⊗ K_{r} ·k(τ) is anisotropic and let r_{0} < r_{1} < … < r_{e} be the elements of J. We know from §13, Theorem 3, that (K_{r_{i}} ·k(τ) | 0 ≤ i ≤ e) is a generic splitting tower of φ ⊗ k(τ) and, since φ⊗k(τ) ∼ ψ⊗k(τ), also of ψ⊗k(τ). Since ψ ≁ 0 we have e ≥ 1. Furthermore φ ⊗ (K_{h−1} · k(τ)) ∼ 0, since φ ⊗ K_{h−1} has kernel form b(τ ⊗ K_{h−1}) for some b ∈ K_{h−1}^{∗} and τ ⊗ k(τ) ∼ 0. Thus h − 1 ∉ J. Therefore

ker(ψ ⊗ K_{r_{e−1}} · k(τ)) = φ_{r_{e−1}} ⊗ K_{r_{e−1}} · k(τ)

has dimension > 2^{n}. This means that ψ ⊗ k(τ) has degree > n.

Let E be a leading field for ψ and ρ the associated leading form of ψ. By §13, Theorem 3, ρ splits over E · k(τ) = E(τ ⊗ E). This implies by §21, Theorem 1, and since dim ρ = dim τ, that ρ ≅ τ ⊗E, and so ψ⊗E ∼ a(τ ⊗E) for some a ∈ E^{∗}. We therefore obtain

φ ⊗ E ∼ τ ⊗ E ⊥ a(τ ⊗ E) = ⟨1, a⟩ ⊗ (τ ⊗ E).

In particular we have deg(φ ⊗ E) = n + 1 or = ∞. We now make use of the lemma below. We apply it to φ and the field extension E of k and obtain τ ⊗ E ∼ 0. But this is nonsense because τ ⊗ E ≅ ρ. Thus we must have deg ψ > n.

Lemma. Continuing with the assumptions of Theorem 3, let L be a field extension of k with deg(φ ⊗ L) > deg φ. Then τ ⊗ L splits.

Proof. Let (K_{r} | 0 ≤ r ≤ h) be the affine standard tower of φ (§13, Definition 4). Thus, K_{0} = k and K_{r+1} = K_{r}(φ_{r}) where φ_{r} = ker(φ ⊗ K_{r}). By §13, Theorem 3, τ ⊗ K_{h−1} · L splits.

Assume for the sake of contradiction that τ ⊗ L is not split. We certainly have h ≥ 2. Let s ∈ [0, h−2] be the largest integer with τ ⊗K_{s} ·L anisotropic.

Let τ̃: = τ ⊗ K_{s} · L and φ̃_{s}: = φ ⊗ K_{s} · L. Then K_{s+1} · L = (K_{s} · L)(φ̃_{s}) and τ̃ ⊗ (K_{s} · L)(φ̃_{s}) = 0. By the usual argument it follows that φ̃_{s} < cτ̃ for some c ∈ (K_{s} · L)^{∗}. In particular, dim φ_{s} ≤ dim τ = dim φ_{h−1}, which is nonsense. Therefore τ ⊗ L must be split.

Remark. We proved the converse of statement (i) in §25, Theorem 3. For characteristic ≠ 2 the converse of statement (ii) of this theorem was also proved in [K_{6}, p. 23 ff.]: If φ ≡ aτ mod J_{n+2}(k), then the (h − 1)-st kernel form φ_{h−1} is defined over k by aτ, φ_{h−1} ≅ (aτ) ⊗ K_{h−1}. So far it has not been possible to extend the proof in [K_{6}] to characteristic 2. One runs into the problem formulated at the end of §24, Part (C), which must be solved at least in a special case.

By means of Theorem 3 we can improve upon Theorem 2:

Theorem 4. Let φ again be a nondegenerate regular form of odd dimension over k, and let ψ be the companion of φ. Let ak^{∗2} be the square class with [a] = QL(φ), in case char k = 2, and ⟨a⟩ = −d(φ), in case char k ≠ 2. Let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower of φ with associated kernel forms φ_{r}: = ker(φ ⊗ K_{r}) and let (L_{s} | 0 ≤ s ≤ e) be a generic splitting tower of ψ with associated kernel forms ψ_{s}: = ker(ψ⊗L_{s}). Assume furthermore that the leading form of φ – therefore also the leading form of ψ – is defined over k by τ. Finally, let h > 1. {Note that otherwise φ ≅ aτ′, ψ ≅ aτ, which is not of interest here.} Then the following statements are equivalent:

(i) dim φ_{h−2} > dim τ + 1. (ii) L_{e−1} ∼_{k} K_{h−1}. (iii) a ∈ D(ψ_{e−1}), and so ψ_{e−1} ≅ (aτ) ⊗ L_{e−1}.

If these statements do not hold, then L_{e−1} ∼_{k} K_{h−2}.

Proof. In §26, Theorem 9, we already determined that there exists a place from L_{e−1} to K_{h−1} over k. Furthermore we have φ ⊥ H ≅ ψ ⊥ [−a], and so φ ⊗ L_{e−1} ∼ ψ_{e−1} ⊥ [−a]. Hence there is certainly a place from K_{h−2} to L_{e−1} over k. This place can be extended to a place from K_{h−1} to L_{e−1} if and only if the form ψ_{e−1} ⊥ [−a] is isotropic, so that a ∈ D(ψ_{e−1}). This establishes the equivalence (ii) ⇔ (iii).

If the form ψ_{e−1} ⊥ [−a] is anisotropic, then it is the kernel form of φ⊗L_{e−1}. If dim φ_{h−2} > dim τ +1, this cannot happen for dimension reasons. Therefore ψ_{e−1} ⊥ [−a] is isotropic in this case. This proves (i) ⇒ (iii).

Now we assume that (i) does not hold, in other words that dim φ_{h−2} = dim τ + 1. We want to show that then L_{e−1} ∼_{k} K_{h−2}. Since K_{h−1} is not equivalent to K_{h−2} over k, this will imply that (iii) does not hold. This will prove the implication (iii) ⇒ (i), in addition to the last statement of the theorem.

We already established above that there is a place from K_{h−2} to L_{e−1} over k. We have to show that there is also a place over k in the reverse direction. By the generic splitting theory this means that the kernel form of ψ ⊗ K_{h−2} must have dimension ≤ 2^{n}.

Let m: = ind(φ ⊗ K_{h−2}). Then

2^{n} + 1 = dim φ_{h−2} = dim φ − 2m.

Furthermore we have ind(ψ ⊗ K_{h−2}) ≥ m. Hence,

dim ker(ψ ⊗ K_{h−2}) ≤ dim ψ − 2m = 2^{n} + 2.

If the inequality is strict, then dim ker(ψ ⊗ K_{h−2}) ≤ 2^{n}, and we are done.

We assume then that the form α: = ker(ψ ⊗ K_{h−2}) has dimension 2^{n} + 2, and want to obtain a contradiction. We have deg α ≥ deg ψ = n, but dim α < 2^{n+1}, since n ≥ 2. This forces deg α = n. By §13, Theorem 3 it is now clear that α has a leading form over K_{h−2}, defined by τ ⊗K_{h−2}. (The form τ ⊗K_{h−2} is in particular anisotropic.) By Theorem 3 this means that

α ∼ (τ ⊗ K_{h−2}) ⊥ χ

for some form χ over K_{h−2} of degree > n. Let τ̃: = τ ⊗ K_{h−2}. We have χ⊗K_{h−2}(τ̃) ∼ α⊗K_{h−2}(τ̃). Therefore α⊗K_{h−2}(τ̃) has degree > n. However, dim α < 2^{n+1}. This forces α ⊗ K_{h−2}(τ̃) ∼ 0. The theory of divisibility by Pfister forms (§21, Theorem 1) then tells us that α ≅ γ ⊗ τ̃ for a suitable form γ over K_{h−2}. Now dim α is in particular divisible by dim τ = 2^{n}. This is the contradiction we were looking for, since dim α = 2^{n} + 2 < 2^{n+1}. This establishes that L_{e−1} is specialization equivalent to K_{h−2} over k.

In the proof of Theorem 4 we did not use Theorem 12 of §26. (Would the proof have been simpler if we had?) If (i) – (iii) in Theorem 4 are valid, this theorem does give us the additional information that the tower (L_{s} | 0 ≤ s ≤ e) is a generic splitting tower of φ, and in particular that h = e.

Chapter IV

Specialization with Respect to Quadratic Places

200 Chapter IV. Specialization with Respect to Quadratic Places

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§28 Quadratic Places; Specialization of Bilinear Forms

In this section we mean by a “form” always a nondegenerate symmetric bilinear form.

To begin with we recall a result from §3. Let λ: K → L ∪ ∞ be a place and 𝔬 = 𝔬_{λ} the associated valuation ring. We defined an additive map

λ_{W}: W (K) ⟶ W (L),

which has the following values for one-dimensional forms (= square classes):

(1) λ_{W} ⟨ε⟩ = ⟨λ(ε)⟩ for ε ∈ 𝔬^{∗},

(2) λ_{W} (α) = 0 for α ∈ Q(K) \ Q(𝔬).

Here we identified the square class group Q(𝔬) = 𝔬^{∗}/𝔬^{∗2} with its image in Q(K) under the natural map ε𝔬^{∗2} ↦ εK^{∗2}.

More generally we have that the natural map {M} ↦ {K ⊗_{𝔬} M} from W (𝔬) to W (K) is injective. (Here {M} denotes the Witt class of a nondegenerate 𝔬-module M). Thus we can also regard W (𝔬) as a subring of W (K). The restriction of λ_{W} to W (𝔬) can then be interpreted as the ring homomorphism W (𝔬) → W (L), induced by the ring homomorphism λ|𝔬: 𝔬 → L. The restriction λ_{W} |W (𝔬) is thus also multiplicative. Furthermore, λ_{W}: W (K) → W (L) is semi-linear with respect to W (𝔬) → W (L). In other words, we have for ξ ∈ W (𝔬) and η ∈ W (K), that

λ_{W} (ξη) = λ_{W} (ξ) · λ_{W} (η).

Starting from the map λ_{W} we developed a specialization theory for bilinear forms in §3. In particular we associated to a form φ with good reduction with respect to λ (i.e., a form over K that comes from a nondegenerate bilinear module over 𝔬) a form λ_{∗}(φ) over L such that {λ_{∗}(φ)} = λ_{W} ({φ}) and dim λ_{∗}(φ) = dim φ. The form λ_{∗}(φ) is only determined up to stable isometry by the form φ however. In truth we thus associated to φ only a stable isometry class of forms over L. Nonetheless we will disregard this fact and just speak of λ_{∗}(φ) as if it were a form over L.

Note that if char L ≠ 2, then stable isometry is the same as isometry by Witt’s Cancellation Theorem and so λ_{∗}(φ) is uniquely determined – up to isomorphism – by λ and φ.

Now we would like to associate to a “quadratic place” Λ: K → L ∪ ∞ (see the following definition) a map Λ_{W}: W (K) → W (L) in a similar manner as was done above and, using this foundation, develop a specialization theory for quadratic places that generalizes the theory developed in §3.

Definition 1. Let K and L be fields. A quadratic place (= Q-place) from K to L is a triple (λ, H, χ) consisting of a place λ: K → L ∪ ∞ (in the usual sense), a subgroup H of Q(K) that contains Q(𝔬) (with 𝔬: = 𝔬_{λ}) and a group homomorphism χ: H → Q(L) with χ(⟨ε⟩) = ⟨λ(ε)⟩ for every ε ∈ 𝔬^{∗}, and thus χ(α) = λ_{∗}(α) = λ_{W} (α) for every α ∈ Q(𝔬).

We will often denote the triple (λ, H, χ) by a Greek capital letter, for instance Λ, and use the symbolic notation Λ: K → L ∪ ∞. We call a group homomorphism from a subgroup H of Q(K) to Q(L) a (Q(L)-valued) character of H.

Example. To every place λ: K → L ∪ ∞ we can associate a quadratic place λ̂: = (λ, H_{λ}, χ_{λ}) by letting H_{λ} = Q(𝔬) and taking for χ_{λ} the character ⟨ε⟩ ↦ ⟨λ(ε)⟩ given by λ. This Q-place λ̂ can then be “extended” to a Q-place Λ: = (λ, H, χ) if we choose a subgroup H of Q(K) which contains Q(𝔬) and somehow continue the character χ_{λ} to a character χ: H → Q(L). Such a continuation is always possible since H and Q(L) are elementary abelian of exponent 2, and can thus be interpreted as 𝔽_{2}-vector spaces.

In this set-up we imagine that λ̂ is the same object as λ, but just dressed up, so that the notion of quadratic place generalizes the notion of place.

Theorem 1. Let Λ = (λ, H, χ) be a Q-place from K to L.

(a) There is precisely one additive map

Λ_{W}: W (K) ⟶ W (L)

with the following properties:

(3) Λ_{W} (α) = χ(α), if α ∈ H.

(4) Λ_{W} (α) = 0, if α ∈ Q(K) \ H.

(b) The restriction of Λ_{W} to the ring W (K, H), generated by H in W (K), is a ring homomorphism from W (K, H) to W (L), and Λ_{W} is semilinear with respect to this ring homomorphism.

Proof. The uniqueness of Λ_{W} is clear since W (K) is additively generated by Q(K). In order to prove the existence, we choose a subgroup H_{0} of H with H = H_{0} × Q(𝔬) (i.e., a “complement” H_{0} of Q(𝔬) in H). Then we define

(5)  Λ_{W} (ξ): = ∑_{α∈H_{0}} χ(α)λ_{W} (αξ)  (ξ ∈ W (K)).

This map Λ_{W}: Q(K) → Q(L) is clearly additive. It satisfies conditions (3) and (4), and also possesses the properties stated in part (b), as can be verified instantly.

Assume now that Λ = (λ, H, χ): K → L∪∞ is a Q-place and that 𝔬: = 𝔬_{λ}. We choose a subgroup H_{0} of H with H: = H_{0} × Q(𝔬). Let φ be a form over K.

§28 Quadratic Places; Specialization of Bilinear Forms 203

Definition 2. We say that φ has good reduction (abbreviated GR) with respect to Λ, when φ has a decomposition (modulo stable isometry)

(6)  φ ≈ ⊥_{α∈H_{0}}αφ_{α},

in which the forms φ_{α} are λ-unimodular. We then also say that φ is Λunimodular. Here we used the notation αφ_{α}: = α ⊗ φ_{α}. (If α = ⟨a⟩, then αφ_{α} = aφ_{α} up to isometry.)

One should note that Λ-unimodularity is independent of the choice of the complement H_{0} of Q(𝔬) in H.

Example. If H = Q(K), then every form φ over K has good reduction with respect to Λ. Namely, if φ ≅ r × ({0 \atop 1} {1 \atop 0}) is hyperbolic, then φ is actually λunimodular. Otherwise we choose a diagonalization φ ≅ ⟨a_{1}, …, a_{n}⟩ and take those elements ⟨a_{j}⟩ ∈ Q(K) together that are in the same coset of Q(𝔬).

Definition 3. Assume that φ has GR with respect to Λ. We choose a decomposition (6) of Λ and let

(7)  Λ_{∗}(φ): = ⊥_{α∈H_{0}}χ(α)λ_{∗}(φ_{α}).

We call Λ_{∗}(φ) a specialization of φ with respect to Λ.

Theorem 2. If φ has GR with respect to Λ then, up to stable isometry, the form Λ_{∗}(φ) does not depend on the choice of the decomposition (6), and also not on the choice of the complement H_{0} of Q(𝔬) in H.

Proof. We clearly have

(8)  Λ_{W} ({φ}) = {Λ_{∗}(φ)}.

Moreover, dim Λ_{∗}(φ) = dim φ. Noting that a form over L is uniquely determined up to stable isometry by its Witt class and dimension finishes the proof.

In the future we will call Λ_{∗}(φ) “the” specialization of φ with respect to Λ. Although sloppy, we hope this notation is harmless and helpful.

Remark. This notion of specialization generalizes the one from §3: if λ is a place from K to L, then a form φ over K has GR with respect to λ̂ if and only if it has GR with respect to λ, and then we have λ̂_{∗}(φ) ≈ λ_{∗}(φ).

Our goal is now to use these notions of good reduction and specialization to build a theory, analogous to the theory developed in §3 and then – in the case char L ≠ 2 – to obtain results about the generic splitting of Λ_{∗}(φ) from the generic splitting behaviour of φ, analogous to §4, see §30 below.

It is obvious that for forms φ, ψ over K which have GR with respect to Λ, also φ ⊥ ψ and φ ⊗ ψ have GR with respect to Λ and that

(9)  Λ_{∗}(φ ⊥ ψ) ≈ Λ_{∗}(φ) ⊥ Λ_{∗}(ψ), Λ_{∗}(φ ⊗ ψ) ≈ Λ_{∗}(φ) ⊗ Λ_{∗}(ψ).

It is not clear however that the analogue of §3, Theorem 4 remains valid. When φ and φ ⊥ ψ have GR with respect to Λ, why should ψ then also have GR with respect to Λ?

§3, Theorem 4 was crucial for the development of the generic splitting theory of §4. In order to save this theorem we will now broaden the notion of “good reduction” to the new notion of “almost good reduction”.

Given a Q-place Λ = (λ, H, χ), we choose a subgroup S of Q(K) with Q(K) = S × H. For every form φ over K we then have a decomposition

(10)  φ ≈ ⊥_{s∈S}sφ_{s}

with Λ-unimodular forms S (even with ≅ instead of ≈). This is obvious for φ ≅ ⟨a⟩ one-dimensional and for φ ≅ ({0 \atop 1} {1 \atop 0}), and so clear for every form φ over K. We call the decomposition (10) a Λ-modular decomposition of φ.

Lemma 1. For every s ∈ S, the Witt class of Λ_{∗}(φ_{s}) in a Λ-modular decomposition (10) of φ is uniquely determined by φ, Λ and s.

Proof. For every s ∈ S we choose a decomposition (see (6) above)

φ_{s} ≈ ⊥_{α∈H_{0}}αφ_{s,α}

with λ-unimodular forms φ_{s,α}. Then

φ ≈ ⊥_{(s,α)∈S×H_{0}}sαφ_{s,α}.

Clearly, λ_{W} ({sαφ}) = {λ_{∗}(φ_{s,α})}. Hence the Witt class of λ_{∗}(φ_{s,α}) is uniquely determined by φ and Λ for fixed s and α. It follows that the Witt class of

Λ_{∗}(φ_{s}) ≈ ⊥_{α∈H_{0}}χ(α)λ_{∗}(φ_{s,α})

is uniquely determined by φ and Λ for fixed s and H_{0}. By Theorem 2, Λ_{∗}(φ_{s}) does not change, up to stable isometry, if we pass from H_{0} to a different complement of Q(𝔬) in H.

Definition 4. Consider a Λ-modular decomposition (11) of φ. We say that φ has almost good reduction (abbreviated AGR) with respect to Λ if all forms Λ_{∗}(φ_{s}) with s ∈ S, s ≠ 1, are metabolic. We then define

Λ_{∗}(φ): = Λ_{∗}(φ_{1}) ⊥ \frac{dim φ − dim φ_{1}}{2} × H̃,

with H̃: = ({0 \atop 1} {1 \atop 0}), and call Λ_{∗}(φ) “the” specialization of φ with respect to Λ..

It is clear from Lemma 1 that the property AGR is independent of the choice of Λ-modular decomposition (10), and that also the Witt class of Λ_{∗}(φ) does not depend on this choice. Since dim Λ_{∗}(φ) = dim φ we have that Λ_{∗}(φ) is uniquely determined by φ up to stable isometry. It is now also clear that the property AGR and the stable isometry class of Λ_{∗}(φ) are independent of the choice of the complement S of H in Q(K).

One should note that for the property AGR all three components λ, H, χ of Λ = (λ, H, χ) play a role, whereas GR only depends on λ and H.

One easily verifies the following

Theorem 3. Let φ and ψ be forms over K. If φ and ψ have almost good reduction with respect to Λ, this is also the case for φ ⊥ ψ and φ ⊗ ψ, and we have

(11)  Λ_{∗}(φ ⊥ ψ) ≈ Λ_{∗}(φ) ⊥ Λ_{∗}(ψ),
(12)  Λ_{∗}(φ ⊗ ψ) ≈ Λ_{∗}(φ) ⊗ Λ_{∗}(ψ).

In addition, the analogue of §3, Theorem 4 also holds for almost good reduction. The proof of this analogue – in the current state of affairs – will turn out to be trivial.

Theorem 4. Let φ and ψ be forms over K. Assume that the forms φ and φ ⊥ ψ have almost good reduction with respect to Λ. Then ψ also has almost good reduction with respect to Λ.

Proof. We choose Λ-modular decompositions

φ ≈ ⊥_{s∈S}sφ_{s}, ψ ≈ ⊥_{s∈S}sψ_{s}.

Then

φ ⊥ ψ ≈ ⊥_{s∈S}s(φ_{s} ⊥ ψ_{s})

is a Λ-modular decomposition of φ ⊥ ψ. For every s ∈ S we have

Λ_{∗}(φ_{s} ⊥ ψ_{s}) ≈ Λ_{∗}(φ_{s}) ⊥ Λ_{∗}(ψ_{s}).

If s ≠ 1, then Λ_{∗}(φ_{s}) ∼ 0 and Λ_{∗}(φ_{s} ⊥ ψ_{s}) ∼ 0, and thus also Λ_{∗}(ψ_{s}) ∼ 0.

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§29 Almost Good Reduction with Respect to Extensions of Quadratic Places

In this section a “form” will again always mean a nondegenerate (symmetric) bilinear form.

Definition 1. Let Λ: K → L ∪ ∞ and Λ′: K → L ∪ ∞ be two quadratic places from K to L, Λ = (λ, H, χ), Λ′ = (λ′, H′, χ′). We call Λ′ an expansion of Λ, and write Λ ⊂ Λ′, when λ = λ′, H ⊂ H′, and χ′|H = χ.

Note that every Q-place Λ = (λ, H, χ) is an expansion of the Q-place λ̂ = (λ, Q(𝔬), λ_{W} |Q(𝔬)) with 𝔬 = 𝔬_{λ}.

Remark. If Λ ⊂ Λ′ and if some form φ over K has good reduction (respectively, almost good reduction) with respect to Λ, then it obviously has GR (respectively, AGR) with respect to Λ′ and we have Λ′_{∗}(φ) = Λ_{∗}(φ).

Consider a Q-place Λ = (λ, H, χ) from K to L. If k is a subfield of K, we obtain from Λ a Q-place Γ = (γ, D, ψ) from k to L as follows: let γ be the restriction λ|k of λ. The group D ⊂ Q(k) is the preimage of H under the homomorphism j: Q(k) → Q(K), which maps a square class ⟨a⟩ = ak^{∗2} of k to the square class ⟨a⟩_{K} = aK^{∗2}. (Note that ⟨a⟩_{K} = ⟨a⟩ ⊗ K when we interpret square classes as one-dimensional forms.) Finally ψ is the character χ ∘ j: D → Q(L).

Definition 2. We call Γ the restriction of the Q-place Λ to k and write Γ = Λ|k.

Caution! If Λ = λ̂ for a place λ: K → L ∪ ∞ and γ = λ|k, then Λ|k ⊃ γ̂, but Λ|k is in general different from γ̂.

The following important theorem for quadratic places has no counterpart for places.

Theorem 1. Let k be a subfield of K and φ a form over k. Let Λ: K → L∪∞ be a Q-place and Γ = Λ|k. The form φ ⊗ K has AGR with respect to Λ if and only if φ has AGR with respect to Γ, and then Λ_{∗}(φ ⊗ K) = Γ_{∗}(φ).

Proof. As above we write Λ = (λ, H, χ), Γ = (γ, D, ψ) with γ = λ|k etc. We choose a subgroup S of Q(k) with Q(k) = S × D. The homomorphism j: Q(k) → Q(K) maps S isomorphically to a subgroup of Q(K), which we denote by S_{K}. We further choose a subgroup T ⊃ S_{K} of Q(K) such that Q(K) = T × H. It is always possible to do this. We now start with a Γ - modular decomposition of φ,

φ ≈ ⊥_{s∈S} s φ_{s},

for Γ -unimodular forms φ_{s} over k. We can interpret

φ ⊗ K ≈ ⊥_{s∈S} s_{K}(φ_{s} ⊗ K)

as a Λ-modular decomposition of φ⊗K for the decomposition Q(K) = T ×H, in which the components for t ∈ T \S_{K} are all zero. By definition φ has AGR with respect to Γ if and only if Γ_{∗}(φ_{s}) ∼ 0 for all s ≠ 1 in S, and φ ⊗ K has AGR with respect to Λ if and only if Λ_{∗}(φ_{s} ⊗ K) ∼ 0 for all s ≠ 1 in S. But for every s ∈ S we have Λ_{∗}(φ_{s} ⊗ K) ≈ Γ_{∗}(φ_{s}). Thus φ ⊗ K has AGR with respect to Λ if and only if φ has AGR with respect to Γ, and in this case we have, with r: = \frac{1}{2}(dim φ − dim φ_{1}), that

Λ_{∗}(φ ⊗ K) = Λ_{∗}(φ_{1} ⊗ K) ⊥ r × H̃ ≈ Γ_{∗}(φ_{1}) ⊥ r × H̃ = Γ_{∗}(φ).

Definition 3. As before, let K be a field and k a subfield of K. Consider Q-places Λ: K → L ∪ ∞ and Γ: k → L ∪ ∞. We call Λ an extension of Γ when Λ|k ⊃ Γ, and a strict extension of Γ when actually Λ|k = Γ.

Theorem 2. Let Λ: K → L ∪ ∞ be an extension of Γ: k → L ∪ ∞. Let φ be a form over k which has AGR with respect to Γ. Then φ ⊗ K has AGR with respect to Λ and Λ_{∗}(φ ⊗ K) = Γ_{∗}(φ).

Proof. By Theorem 1 the statement holds when Λ is a strict extension of Γ. Thus it suffices to consider the case where K = k and Λ = (λ, H, χ) is an expansion of Γ = (γ, D, ψ). Thus, let K = k, D ⊂ H, χ|D = ψ.

We choose subgroups S_{1} of H and T of Q(k) with H = S_{1} × D, Q(k) = T × H. Then S: = T × S_{1} is a subgroup of Q(k) with Q(k) = S × D. Let

φ ≈ ⊥_{(α,β)∈T ×S_{1}} αβ φ_{α,β}

be a Γ -modular decomposition of φ. We let

φ_{α}: = ⊥_{β∈S_{1}} β φ_{α,β} (α ∈ T).

For every α ∈ T the form φ_{α} is Λ-unimodular and

(1)  Λ_{∗}(φ_{α}) ≈ ⊥_{β∈S_{1}} χ(β) Γ_{∗}(φ_{α,β}).

It follows that

φ ≈ ⊥_{α∈T} α φ_{α}

§29 Almost Good Reduction with Respect to Extensions of Q-Places 209

is a Λ-modular decomposition of φ. If α ≠ 1, then αβ ≠ 1 for every β ∈ S_{1}, and so Γ_{∗}(φ_{αβ}) ∼ 0 for every β ∈ S_{1}. From (1) we then see that Λ_{∗}(φ_{α}) ∼ 0. Therefore φ has AGR with respect to Λ.

Furthermore,

(2)  Λ_{∗}(φ) ≈ Λ_{∗}(φ_{1}) ⊥ \frac{dim φ − dim φ_{1}}{2} × H̃.

If β ∈ S_{1}, β ≠ 1, then Γ_{∗}(φ_{1,β}) ∼ 0, and we see from the expression (1) with α = 1 that

(3)  Λ_{∗}(φ_{1}) ≈ Γ_{∗}(φ_{1,1}) ⊥ \frac{dim φ_{1} − dim φ_{1,1}}{2} × H̃.

From (2) and (3) we obtain

Λ_{∗}(φ) ≈ Γ_{∗}(φ_{1,1}) ⊥ \frac{dim φ − dim φ_{1,1}}{2} × H̃ ≈ Γ_{∗}(φ).

Theorem 3. Let Γ: k → L ∪ ∞ be a Q-place and φ a form over k which has AGR with respect to Γ. Assume that there exists a field extension K ⊃ k such that φ ⊗ K is isotropic and Γ has an extension to a Q-place Λ: K → L ∪ ∞. Then Γ_{∗}(φ) is isotropic. More precisely,

ind Γ_{∗}(φ) ≥ ind(φ ⊗ K).

Proof. We have φ⊗K ≈ ψ_{0} ⊥ r×H̃ with ψ_{0} anisotropic, r = ind(φ⊗K) > 0. By Theorem 2, φ ⊗ K has AGR with respect to Λ and Γ_{∗}(φ) ≈ Λ_{∗}(φ ⊗ K). Theorem 4 from §28 tells us that ψ_{0} has AGR with respect to Λ. Therefore

Γ_{∗}(φ) ≈ Λ_{∗}(ψ_{0}) ⊥ r × H̃.

Theorem 3 inspires the hope that for almost good reduction there are theorems about the generic splitting of Γ_{∗}(φ), analogous to those in §4. As in §4 one will have to assume that the characteristic of L is different from two though.

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§30 Realization of Quadratic Places; Generic Splitting of Specialized Forms in Characteristic ≠ 2

We consider fields of arbitrary characteristic and want to investigate “realizations” of a quadratic place in the sense of the following definition.

Definition 1. Let Λ: K → L ∪ ∞ be a quadratic place. A realization of Λ (as an ordinary place) is a pair (i, μ) consisting of a field extension i: K ↪ E and a place μ: E → L ∪ ∞ with the property Λ ⊂ μ̂|K. {Here μ̂ denotes the quadratic place determined by μ, cf. §28.} The realization is called strict when actually Λ = μ̂|K.

Strict realizations are not easy to find, but in general the following theorem holds which suffices for our generic splitting theory.

Theorem 1. For every quadratic place Λ: K → L ∪ ∞ there exists a realization (i: K ↪ E, μ) with E purely transcendental over K.

For the proof we need a lemma, which follows easily from general principles pertaining to the extension of valuations, cf. [Bo_{2}, §2, n^{∘} 4, Prop.3 and §8, n^{∘} 3, Th.1].

Lemma 1. Let K be a quadratic field extension of a field E, K = E(α) with α^{2} = a ∈ E, α ∉ E. Let ρ: E → L ∪ ∞ be a place with ρ(ac^{2}) = 0 or ∞ for all c ∈ E. Then ρ extends uniquely to a place λ: K → L ∪ ∞.

Proof of Theorem 1. Let Λ = (λ, H, χ), 𝔬: = 𝔬_{λ} and 𝔪 the maximal ideal of the valuation ring 𝔬. We choose a subgroup P of Q(K) with Q(K) = P ×Q(𝔬) and a family (π_{i} | i ∈ I) in 𝔪 such that the square classes ⟨π_{i}⟩, i ∈ I, form a basis of the 𝔽_{2}-vector space P. Finally we choose families of indeterminates (t_{i} | i ∈ I) and (u_{i} | i ∈ I) over the fields K and L respectively.

Let E′: = K(t_{i} | i ∈ I) and F: = L(u_{i} | i ∈ I). The place λ: K → L ∪ ∞ extends uniquely to a place λ̃: E′ → F ∪∞ with λ̃(u_{i}) = t_{i} for every i ∈ I, cf. [Bo_{2}, §10, Prop.2]. {Note: with 𝔬̃: = 𝔬λ̃ and 𝔪̃ the maximal ideal of 𝔬̃ we have 𝔬̃/𝔪̃ ≅ (𝔬/𝔪)(\overline{t_{i}} | i ∈ I), where \overline{t_{i}} denotes the image of t_{i} in 𝔬̃/𝔪̃. The family (\overline{t_{i}} | i ∈ I) is algebraically free over 𝔬/𝔪. The natural map K^{∗}/𝔬^{∗} → E^{′∗}/𝔬̃^{∗} is an isomorphism.}

For every i ∈ I and f ∈ E′ we have λ̃(π_{i}f^{2}) = 0 or ∞. Let E: =

E′(\sqrt{π_{i}t_{i}^{−1}} | i ∈ I). This field is again purely transcendental over K. It

follows from Lemma 1 (possibly with Zorn’s lemma) that λ̃ extends uniquely to a place ρ: E → F ∪ ∞. (We have ρ(\sqrt{π_{i}t_{i}^{−1}}) = 0 for every i ∈ I.) Now ⟨π_{i}⟩_{E} = ⟨t_{i}⟩_{E}. Thus ⟨π_{i}⟩_{E} has GR with respect to ρ and ρ_{∗}(⟨π_{i}⟩_{E}) = ⟨u_{i}⟩ for every i ∈ I.

Finally we choose for every i ∈ I an element a_{i} of L with χ(⟨π_{i}⟩) = ⟨a_{i}⟩. There exists a place σ: F → L∪∞ over L, in general not uniquely determined, with σ(u_{i}) = a_{i} for every i. For every such place σ we have σ̂ ∘ ρ |K ⊃ Λ, which can be verified easily.

In what follows we require that all fields have characteristic ≠ 2. As before, a “form” is a nondegenerate (symmetric) bilinear form, which is now the same as a nondegenerate quadratic form.

As indicated at the end of §29, we want to study the generic splitting of quadratically specialized forms Λ_{∗}(φ) of forms with almost good reduction in a manner, similar to what was done in §4 for specialized forms λ_{∗}(φ) of forms with good reduction. For this purpose we need a supplement to Theorem 2 in §4 (which itself is a special case of §9, Theorem 4) about generic zero fields.

Theorem 3 [K_{5}, Th.3.3.ii]. Let γ: k → L ∪ ∞ be a place and φ a form over k with bad (= not good) reduction with respect to γ. Assume there exists an element c ∈ D(φ) with γ(c) ≠ 0, ∞. Then γ extends to a place λ: k(φ) → L ∪ ∞.

Proof. We may assume that φ = ⟨a_{1}, …, a_{n}⟩ with n ≥ 2, γ(a_{i}) ≠ ∞ for all i, γ(a_{1}) ≠ 0, γ(a_{n}c^{2}) = 0 or ∞ for every c ∈ k. We can write k(φ) =

k(x_{1}, …, x_{n}) with the only relation ∑_{1}^{n} a_{i}x_{i}^{2} = 0. The elements x_{1}, …, x_{n−1}

are algebraically independent over k and k(φ) is a quadratic extension E(x_{n}) of the purely transcendental extension E = k(x_{1}, …, x_{n−1}) of k.

Let u_{1}, …, u_{n−1} be indeterminates over L and F: = L(u_{1}, …, u_{n−1}). Then γ has a unique extension μ: E → F with μ(x_{i}) = u_{i} (i = 1, …, n − 1). We want to show that μ(x_{n}^{2}z^{2}) = 0 or ∞ for every z ∈ E. By Lemma 1 above we then know that μ extends to a place ρ: k(φ) → F ∪ ∞. By composing ρ with a place σ: F → L∪∞ over L, we obtain a place λ = σ ∘ρ: k(φ) → L∪∞, which extends γ.

Let z ∈ E^{∗}. We write z = dfg^{−1} with d ∈ k and polynomials f, g ∈ k[x_{1}, …, x_{n−1}] whose coefficients are all in the valuation ring 𝔬 of γ, but not all in the maximal ideal of 𝔬. Then μ(f) and μ(g) are finite and nonzero. We obtain

μ(x_{n}^{2}z^{2}) = −[γ(a_{1})u_{1}^{2} + … + γ(a_{n−1})u_{n−1}^{2}]μ(f)^{2}μ(g)^{−2}γ(a_{n}^{−1}d^{2}).

On the right hand side all factors, except the last one, are finite and nonzero. In contrast, γ(a_{n}^{−1}d^{2}) = 0 or ∞. Therefore the same is true for μ(x_{n}^{2}z^{2}).

Theorem 4. Let γ: k → L ∪ ∞ be a place and φ a form over k which has AGR with respect to γ̂. Assume that the form γ̂_{∗}(φ) is isotropic. Then there exists a place λ: k(φ) → L ∪ ∞, which extends γ.

§30 Realization of Quadratic Places … 213

Proof. Let Γ: = γ̂. If φ has GR with respect to γ then our statement is covered by Theorem 2 from §4, since γ̂_{∗}(φ) = γ_{∗}(φ) in this case. Assume now that φ has bad reduction with respect to γ. We choose a subgroup P of Q(K) with Q(k) = Q(𝔬_{γ}) × P. Let

φ ≅ ⊥_{α∈P} α φ_{α}

be a Γ -modular decomposition of φ. This implies that all forms φ_{α} are γunimodular. If φ_{1} ≠ 0 then Theorem 3 tells us that γ extends to a place from k(φ) to L.

Finally assume that φ_{1} = 0. We have dim φ = dim Γ_{∗}(φ) ≥ 2. Thus there certainly exists an α_{0} ∈ P with φ_{α_{0}} ≠ 0. Since φ has AGR, we have γ_{∗}(φ_{α_{0}}) ∼ 0. The form φ̃: = α_{0}φ has AGR with respect to Γ, since φ_{1} = 0 and thus γ_{∗}(φ_{1}) = 0. (γ_{∗}(φ_{1}) ∼ 0 would suffice.) We have k(φ) = k(φ̃). Γ_{∗}(φ̃) has an orthogonal summand γ_{∗}(φ_{α_{0}}) and is thus isotropic. By the above γ extends to a place from k(φ̃) to L.

Now we can prove a theorem, crucial for a generic splitting theory of specialized forms. It is the converse of §29, Theorem 3 in characteristic ≠ 2.

Theorem 5. Let Γ: k → L ∪ ∞ be a Q-place and φ a form over k which has AGR with respect to Γ. Assume that the form Γ_{∗}(φ) is isotropic. Let K ⊃ k be a generic zero field. Then there exists a Q-place Λ: K → L ∪ ∞, which extends Γ {i.e., with Λ|k ⊃ Γ, cf. §29, Definition 3}.

Proof. Following Theorem 1 we choose a purely transcendental extension E ⊃ k together with a place μ: E → L∪∞ such that μ̂|k ⊃ Γ. By §29, Theorem 2, φ⊗E has AGR with respect to μ̂ and μ̂_{∗}(φ⊗E) = Γ_{∗}(φ). Therefore μ̂_{∗}(φ⊗E) is isotropic. The free compositum K · E of K and E over k is specialization equivalent to k(φ) · E = E(φ ⊗ E) over E. By Theorem 4 we can extend μ to a place ρ: K · E → L ∪ ∞. Let Λ: K → L ∪ ∞ be the restriction of the Q-place ρ̂ to K. Clearly,

Λ|k = (ρ̂|K)|k = ρ̂|k = (ρ̂|E)|k,

and also ρ̂|E ⊃ μ̂. Therefore

Λ|k ⊃ μ̂|k ⊃ Γ.

With this theorem and §29, Theorem 3, we have all the ingredients at our disposal to replicate the arguments of §4 about generic splitting of a specialized form. If we traverse the proof of §4, Theorem 3 and replace the words “place” and “good reduction” everywhere by “Q-place” and “almost good reduction” respectively, we obtain the following theorem about the Witt decomposition of a specialized form.

Theorem 6. Let φ be a form over a field k. Let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower of φ with associated higher kernel forms φ_{r} and indices i_{r}. Let Γ: k → L∪∞ be a Q-place, with respect to which φ has almost good reduction.

Finally, consider a Q-place Λ: K_{m} → L ∪ ∞ for some m, 0 ≤ m ≤ h, which extends Γ and which in the case m < h cannot be extended to K_{m+1}. Then φ_{m} has good reduction with respect to Λ. The form Γ_{∗}(φ) has kernel form Λ_{∗}(φ) and Witt index i_{0} + … + i_{m}.

A subtle point to make here is that φ_{m} has GR with respect to Λ, not just AGR. Namely, if φ_{m} only had AGR, then Λ_{∗}(φ_{m}) would certainly be isotropic.

We now also obtain a detailed proposition about the generic splitting of Γ_{∗}(φ), which parallels word for word Scholium 3 of §4, and whose formulation we may thus leave to the Reader.

§31 Stably Conservative Reduction of Quadratic Forms

The splitting theory of a quadratically specialized form, developed above, required the assumption that the characteristic is different from two. We would like to establish such a theory also for characteristic two, generalizing our theory in Chapter I, from §5 onwards, and in Chapter II.

Thence we will consider quadratic forms instead of bilinear forms. In the process we will endeavour to stay as close as possible to the specialization concept of “almost good reduction”, developed in §28–30. Nevertheless important new problems will occur, which all arise from the lack of obedience of many forms and the existence of quasilinear parts. An obvious, immediate generalization of the concept of “almost good reduction” will not suffice.

We first recall some definitions from §7 and §11.

Let λ: K → L ∪ ∞ be a place between fields of arbitrary characteristic, 𝔬 = 𝔬_{λ}, 𝔪 = 𝔪_{λ}, k = 𝔬/𝔪. The place λ yields a field embedding \overline{λ}: k → L.

We choose a subgroup T of Q(K) with Q(K) = T × Q(𝔬). A quadratic space (E, q) = E over K is called weakly obedient with respect to λ if it has a decomposition

(1)  E = ⊥_{α∈T} F_{α}, F_{α} ≅ ⟨α⟩ ⊗ E_{α}

in which every E_{α} is a quadratic space over K of the form

(2)  E_{α} = K ⊗_{𝔬} M_{α} = KM_{α},

for some quadratic module M_{α} over 𝔬, which is free of finite rank, and for which the quadratic module M_{α}/𝔪M_{α} over k is nondegenerate. (M_{α} is then called “reduced nondegenerate”.) We also called (1) a weakly λ-modular decomposition of E. We described spaces of the form (2) as having fair reduction (FR), and denoted the space M_{α}/𝔪M_{α} ⊗\overline{λ} L by λ_{∗}(E_{α}). We saw in §11 that for every α the Witt class {λ_{∗}(E_{α})} is determined solely by E, λ and α.

We should keep in mind that the quasilinear part QL(M_{α}/𝔪M_{α}) of M_{α}/𝔪M_{α} has to be anisotropic, but that QL(λ_{∗}(E_{α})) could be isotropic. {We defined Witt equivalence over fields also for forms with isotropic quasilinear part, cf. §6.}

If E actually has the decomposition (1), (2), in which every M_{α} is a nondegenerate quadratic 𝔬-module, then we said that E is obedient with respect to λ, and further that E_{α} has good reduction (GR) with respect to Λ. Finally we then called (1) a λ-modular decomposition of E.

Assume now that λ is enlarged to a quadratic place Λ = (λ, H, χ): K → L ∪ ∞. We choose direct decompositions H = H_{0} × Q(𝔬), Q(K) = S × H. The group T from above is now S × H_{0} = SH_{0}.

We switch to the language of quadratic forms instead of quadratic modules, and thus use φ, φ_{α} etc. instead of E, E_{α} etc. A “form” will now always be a quadratic form.

Let φ be a nondegenerate form over K.

Definition 1. We say that φ has fair reduction (= FR) with respect to Λ, or that φ is weakly Λ-unimodular, if φ has a decomposition

(3)  φ ≅ ⊥_{α∈H_{0}}α ⊗ φ_{α} = ⊥_{α∈H_{α}}αφ_{α}

in which every φ_{α} has FR with respect to λ, and then set

(4)  Λ_{∗}(φ): = ⊥_{α∈H_{0}}χ(α)λ_{∗}(φ_{α}).

If there actually exists a decomposition (3) in which all φ_{α} have GR with respect to λ, then we say that φ has GR with respect to Λ, or that φ is Λ-unimodular.

As in §28 we see that Λ_{∗}(φ) is determined by φ up to isometry, and also that it is independent of the choice of the complement H_{0} of Q(𝔬) in H, if we utilize the weak specialization λ_{W} from §11 instead of the map λ_{W} from §3.^{41} {In the case of good reduction, the operator λ_{W} from §7 suffices.}

It goes without saying that φ must be weakly obedient (resp. obedient) with respect to λ in order for φ to have FR (resp. GR) with respect to Λ. It can happen that Λ_{∗}(φ) is degenerate; for fair reduction even in the case where λ is the canonical place K → 𝔬/𝔪 ∪ ∞ associated to 𝔬.

Remarks.

(1) If φ, ψ have FR (resp. GR) with respect to Λ, and if φ is strictly regular, then φ ⊥ ψ has FR (resp. GR) with respect to Λ and

(5)  Λ_{∗}(φ ⊥ ψ) ≅ Λ_{∗}(φ) ⊥ Λ_{∗}(ψ),

which can be seen easily.

(2) If σ is a bilinear form which has GR with respect to Λ, φ is a quadratic form which has FR (resp. GR) with respect to Λ, and if Λ_{∗}(φ) is strictly regular, then σ ⊗ φ has FR (resp. GR) with respect to Λ and

(6)  Λ_{∗}(σ ⊗ φ) ≅ Λ_{∗}(σ) ⊗ Λ_{∗}(φ),

which can also be verified easily. Note that Λ_{∗}(σ) is determined by Λ, σ only up to stable isometry. Nevertheless this formula (with true isometry) is correct.

41In order to be able to repeat the conclusion of §28, it is important that for our definition of Witt equivalence over fields (§6, Def. 10), the statement “If E ∼ F, dim E = dim F, then E ≅ F ” also holds for degenerate spaces.

§31 Stably Conservative Reduction of Quadratic Forms 217

(3) It is important that Λ_{∗}(φ) is strictly regular in formula (6), at any rate when Λ_{∗}(σ) is isotropic. This follows already from the following observation. Let char L = 2 and let a, b, c ∈ L^{∗}. Then

⟨a, a⟩ ⊗ [c] ≅ [ac, ac] ≅ [ac, 0],

and likewise ⟨b, b⟩ ⊗ [c] ≅ [bc, 0]. Furthermore

⟨a, a⟩ ≅ ⟨{a \atop 1} {1 \atop 0}⟩ ≈ ⟨{b \atop 1} {1 \atop 0}⟩ ≅ ⟨b, b⟩,

but in general [ac, 0] ≅ [bc, 0].

If φ is weakly obedient with respect to λ and if

(1)′  φ = ⊥_{(s,β)∈S×H_{0}}sβ φ_{s,β}

is a weakly λ-unimodular decomposition of φ, in which thus every form φ_{s,β} has FR with respect to λ, then we can combine the forms φ_{s,β}, for fixed s, in a form

(5)  φ_{s}: = ⊥_{β∈H_{0}}β φ_{s,β}

which is weakly Λ-unimodular. This results in a coarser decomposition

(6)  φ = ⊥_{s∈S}s φ_{s}.

We call such a decomposition (6) a weakly Λ-modular decomposition of φ. If φ is actually obedient with respect to λ, then we have such decompositions (5), (6), in which the φ_{s,β} have GR with respect to λ and the φ_{s} have GR with respect to Λ, and then call (6) a Λ-modular decomposition of φ.

Definition 2. (a) We say that φ has almost fair reduction (= AFR) with respect to Λ if φ is weakly obedient with respect to λ, and if in a Λ-modular decomposition (6) of φ the specializations Λ_{∗}(φ_{s}), with s ∈ S \ {1}, are all hyperbolic.

(b) We say that φ has almost good reduction (= AGR) with respect to Λ if there is a Λ-modular decomposition (6) in which Λ_{∗}(φ_{s}) is hyperbolic for each s ∈ S \ {1}.

Note that these properties are independent of the choice of decomposition (6) since every Λ_{∗}(φ_{s}) is up to Witt equivalence uniquely determined by φ, Λ and s.

As in §28 we see that by almost fair reduction the form

Λ_{∗}(φ): = Λ_{∗}(φ_{1}) ⊥ \frac{dim φ − dim φ_{1}}{2} × H

is determined up to isometry (instead of only Witt equivalence) by Λ and φ alone. We call Λ_{∗}(φ) the specialization of φ with respect to Λ.

Next there follow two theorems which can be proved in exactly the same way as the analogous statements in §28 and §29.

Theorem 1. Let φ and ψ be quadratic forms over K. Assume that φ is strictly regular and has AGR with respect to Λ. Assume that ψ has AFR (resp. AGR) with respect to Λ. Then φ ⊥ ψ has AFR (resp. AGR) with respect to Λ and

Λ_{∗}(φ ⊥ ψ) ≅ Λ_{∗}(φ) ⊥ Λ_{∗}(ψ).

Theorem 2. Let φ be a quadratic form, weakly obedient (resp. obedient) with respect to λ: K → L ∪ ∞. As before, let Λ = (λ, χ, H): K → L ∪ ∞ be a Qplace with first component λ. Let further E ⊃ K be a field extension and M: E → L ∪ ∞ a Q-place with M|K ⊃ Λ.

(a) If φ has AFR (resp. AGR) with respect to Λ, then φ ⊗ E has AFR (resp. AGR) with respect to M, and M_{∗}(φ ⊗ E) = Λ_{∗}(φ).

(b) If actually M|K = Λ, and if φ ⊗ E has AFR (resp. AGR) with respect to M, then φ has AFR (resp. AGR) with respect to Λ.

However, can we establish, as in §28, wether for two forms φ, ψ over K where φ and φ ⊥ ψ have AGR with respect to Λ: K → L ∪ ∞, also ψ has AGR with respect to Λ, even when φ and ψ are both strictly regular?

This is a debatable question since (at least for the author) there is no discernible reason why ψ should be obedient with respect to the place λ.

Thus we are forced to make a new turn and follow a different path. The fact that for almost fair reduction, or even almost good reduction, the quasilinear part QL(Λ_{∗}(φ)) could be isotropic, will be an obstacle. We want to exclude this possibility.

Definition 3. Let Λ: K → L ∪ ∞ be a quadratic place and let φ be a form over K. We say that φ has conservative reduction (= CR) with respect to Λ if φ has AFR with respect to Λ and QL(Λ_{∗}(φ)) is anisotropic.

If φ actually has AGR with respect to Λ, then clearly QL(Λ_{∗}(φ)) = Λ_{∗}(QL(φ)). Hence φ certainly has CR in this case if φ is in addition regular. If φ only has AFR, then QL(Λ_{∗}(φ)) may be isotropic, even if φ is regular.

As a result of the following lemma we obtain a connection between conservative reduction with respect to Λ and fair reduction with respect to an ordinary place. In the process we will be able to utilize results from §11.

Lemma 1. Let (K ↪ E, μ) be a realization of Λ, i.e., E ⊃ K is a field extension and μ: E → L ∪ ∞ is a place with μ̂|K ⊃ Λ. {Note that there exists a realization of Λ with E ⊃ K even purely transcendental, as demonstrated in §30.} If φ has conservative reduction with respect to Λ, then φ ⊗ E has fair reduction with respect to μ and μ_{∗}(φ ⊗ E) = Λ_{∗}(φ).

Proof. We use Theorem 2(b). Let φ = ⊥_{s∈S}⟨s⟩ ⊗ φ_{s} be a weakly Λ-modular

decomposition of φ. For every s ∈ S we have ⟨s⟩_{E} ∈ Q(𝔬_{μ}). If s ≠ 1, then φ_{s} ⊗ E has FR with respect to μ, and μ_{∗}(φ_{s} ⊗ E) = Λ_{∗}(φ_{s}) is hyperbolic. Furthermore, the anisotropy of QL(Λ_{∗}(φ)) = QL(Λ_{∗}(φ_{1})) yields that φ_{1} ⊗ E has fair reduction with respect to μ and μ_{∗}(φ_{1} ⊗ E) = Λ_{∗}(φ_{1}). Hence φ ⊗ E has FR with respect to μ and

μ_{∗}(φ ⊗ E) = Λ_{∗}(φ_{1}) ⊥ \frac{dim φ − dim φ_{1}}{2} × H = Λ_{∗}(φ).

Theorem 3. Let φ, ψ be quadratic forms over K, both having CR with respect to Λ: K → L ∪ ∞. Assume that φ < ψ. Then Λ_{∗}(φ) < Λ_{∗}(ψ).

Proof. We choose a place μ: E → L ∪ ∞ with E ⊃ K and μ̂|K ⊃ Λ. By the lemma φ ⊗ E and ψ ⊗ E both have FR with respect to μ with Λ_{∗}(φ) = μ_{∗}(φ ⊗ E), Λ_{∗}(ψ) = μ_{∗}(ψ ⊗ E). The form φ ⊗ E is a subform of ψ ⊗ E. By §11, Theorem 2 (Corollary 2) μ_{∗}(φ ⊗ E) is a subform of μ_{∗}(ψ ⊗ E).

Now we come to the key definition, which allows us to tame disobedient forms at least a little bit.

Definition 4. Let φ be a quadratic form over K and, as before, Λ: K → L∪∞ a quadratic place.

(a) φ has stably conservative reduction (= SCR) with respect to Λ if there exists an r ∈ ℕ_{0} such that φ ⊥ r × H has conservative reduction with respect to Λ.

(b) In this case r × H is a subform of Λ_{∗}(φ ⊥ r × H) by Theorem 3. We then denote by Λ_{∗}(φ) the form over L, uniquely determined up to isometry, for which

Λ_{∗}(φ ⊥ r × H) = Λ_{∗}(φ) ⊥ r × H,

and call Λ_{∗}(φ) the specialization of φ with respect to Λ.

Remark. Λ_{∗}(φ) is independent of the choice of r. Namely, if s ∈ ℕ is arbitrary then, in the situation of Definition 4(a), φ ⊥ (r+s)×H = (φ ⊥ r×H) ⊥ s×H also has conservative reduction and

Λ_{∗}(φ ⊥ (r × s) × H) = Λ_{∗}(φ ⊥ r × H) ⊥ s × H = Λ_{∗}(φ) ⊥ (r + s) × H.

Definition 5. Similarly we define: a quadratic form φ over K has stably almost good reduction (= SAGR) with respect to Λ if there exists an r ∈ ℕ_{0} such that φ ⊥ r × H has AGR with respect to Λ.

Remark. If φ has SAGR, then QL(φ) has good reduction with respect to Λ. Such a form φ has SCR with respect to Λ if and only if Λ_{∗}(QL(φ)) is anisotropic, and then Λ_{∗}(QL(φ)) = QL(Λ_{∗}(φ)).

Finally we can derive a full analogue of the additive statements in §28, Theorem 3 and §28, Theorem 4.

Theorem 4. Let φ and ψ be quadratic forms over K. Assume that φ is strictly regular and that it has SAGR with respect to Λ: K → L ∪ ∞.

(a) If ψ has SCR with respect to Λ, then φ ⊥ ψ has SCR with respect to Λ

and

Λ_{∗}(φ ⊥ ψ) ≅ Λ_{∗}(φ) ⊥ Λ_{∗}(ψ).

(b) If φ ⊥ ψ has SCR with respect to Λ, then ψ has SCR with respect to Λ.

Proof. (a) Choose r ∈ ℕ_{0}, s ∈ ℕ_{0} such that φ ⊥ r × H has AGR and ψ ⊥ s×H has CR with respect to Λ. From Theorem 1 it follows immediately that φ ⊥ ψ ⊥ (r + s) × H has CR with respect to Λ and that

Λ_{∗}(φ) ⊥ r × H ⊥ Λ_{∗}(ψ) ⊥ s × H = Λ_{∗}(φ ⊥ ψ ⊥ (r + s) × H).

Therefore,

Λ_{∗}(φ) ⊥ Λ_{∗}(ψ) = Λ_{∗}(φ ⊥ ψ).

(b) We choose integers r, t ∈ ℕ_{0} with t ≥ r and such that ρ: = φ ⊥ r × H has AGR and that τ: = (φ ⊥ ψ) ⊥ t × H has AFR. We have

ρ ⊥ ψ ⊥ (t − r) × H ≅ τ.

Therefore,

ψ ⊥ (t − r + dim ρ) × H ≅ (−ρ) ⊥ τ.

By part (a) of the Theorem, (−ρ) ⊥ τ has SCR with respect to Λ. Thus ψ has SCR with respect to Λ.

Furthermore we can extend Theorem 3 and part of Theorem 2 to stably conservative reduction.

Theorem 5. Let φ, ψ be quadratic forms over K which have SCR with respect to Λ. Assume that φ < ψ. Then Λ_{∗}(φ) < Λ_{∗}(ψ).

Proof. We choose an r ∈ ℕ such that φ ⊥ r × H and ψ ⊥ r × H have CR. By Theorem 3 we have

Λ_{∗}(φ) ⊥ r × H = Λ_{∗}(φ ⊥ r × H) < Λ_{∗}(ψ ⊥ r × H) = Λ_{∗}(ψ) ⊥ r × H.

Hence, Λ_{∗}(φ) < Λ_{∗}(ψ).

Theorem 6. Let K ⊂ E be a field extension and let Λ: K → L ∪ ∞, M: E → L ∪ ∞ be quadratic places with M|K ⊃ Λ. Let φ be a quadratic form over K which has SCR with respect to Λ. Then φ ⊗ E has SCR with respect to M and M_{∗}(φ ⊗ E) = Λ_{∗}(φ).

Proof. We choose an r ∈ ℕ_{0} such that φ ⊥ r × H has CR with respect to Λ. By Theorem 2(a), φ_{E} ⊥ r × H has AFR with respect to Λ and

M_{∗}(φ_{E} ⊥ r × H) ≅ Λ_{∗}(φ ⊥ r × H) = Λ_{∗}(φ) ⊥ r × H.

In particular, QL(M_{∗}(φ_{E} ⊥ r × H)) is anisotropic. Thus φ_{E} ⊥ r × H has CR with respect to M. Therefore φ_{E} has SCR with respect to Λ and

M_{∗}(φ_{E}) ⊥ r × H ≅ Λ_{∗}(φ) ⊥ r × H.

It follows that M_{∗}(φ_{E}) ≅ Λ_{∗}(φ).

§32 Generic Splitting of Stably Conservative Specialized Quadratic Forms

For a quadratic form φ over a field k, which has SCR with respect to a quadratic place Γ: k → L ∪ ∞, we will show how the splitting of Γ_{∗}(φ) with respect to extensions of the field L, conservative for Γ_{∗}(φ), is regulated by an arbitrarily given generic splitting tower (K_{r} | 0 ≤ r ≤ h) and quadratic places coming from the fields K_{r} (see Theorem 2 below).

For this purpose we need the concept of composition of two quadratic places.

Definition 1. Let Λ = (λ, H, χ): K → L∪∞ and M = (μ, D, ψ): L → F ∪∞ be two quadratic places. Let

H_{0}: = {x ∈ H, χ(α) ∈ D}.

Clearly

M ∘ Λ: = (μ ∘ λ, H_{0}, ψ ∘ (χ|H_{0}))

is a quadratic place from K to F, called the compositum of Λ and M.

The theory of these composita is fraught with danger. We will not look at its details, as we would like to advance as quickly as possible towards the core of a generic splitting theory of stably conservative specialized forms Λ_{∗}(φ). This is possible just with Definition 1 and §31, although many things would become clearer conceptually after a detailed study of the formal properties of composita of quadratic places.

We only note the following trivial fact.

Remark. If Λ′: K → L∪∞, M′: L → F ∪∞ are quadratic places with Λ′ ⊃ Λ, M′ ⊃ M, then M′ ∘ Λ′ ⊃ M ∘ Λ.

Theorem 1. Let Γ: k → L ∪ ∞ be a quadratic place and φ a quadratic form over k which has SCR with respect to Γ. Let r ∈ ℕ_{0} and let K ⊃ k be a field extension which is φ-conservative and generic for the splitting off of r hyperbolic planes. Then the following statements are equivalent:

(1) ind(Γ_{∗}(φ)) ≥ r.

(2) There exists a quadratic place Λ: K → L ∪ ∞ with Λ_{∗}|k ⊃ Γ.

Proof. (2) ⇒ (1): By §31, Theorem 6, φ ⊗ K has SCR with respect to Λ and Λ_{∗}(φ ⊗ K) ≅ Γ_{∗}(φ). Now r × H < φ ⊗ K and thus, by §31, Theorem 5, also r × H < Λ_{∗}(φ ⊗ K). Hence Γ_{∗}(φ) has Witt index ≥ r.

(1) ⇒ (2): We carry out the proof in two steps.

(a) We choose a purely transcendental extension E ⊃ k and a place μ: E → L ∪ ∞ with μ̂|k ⊃ Λ. This is possible by §30, Theorem 1. Then we choose a generic splitting field F ⊃ E of φ ⊗ E for the splitting off of r hyperbolic planes. Finally we choose an s ∈ ℕ_{0} such that φ ⊥ s × H has CR with respect to Γ. By §31, Lemma 1, (φ ⊥ s × H)_{E} = φ_{E} ⊥ s × H has FR with respect to μ, and

μ_{∗}(φ_{E} ⊥ s × H) = Γ_{∗}(φ ⊥ s × H) = Γ_{∗}(φ) ⊥ s × H.

E is a generic splitting field of φ_{E} ⊥ s × H for the splitting off of r + s hyperbolic planes. The form Γ_{∗}(φ) ⊥ s × H over L is nondegenerate since by assumption (1) its quasilinear part is anisotropic. Moreover, this form has Witt index ≥ r + s by assumption (1). Thus, by the generic splitting theory of §12, there exists a place σ: F → L∪∞ which extends μ. For the associated quadratic place σ̂: F → L∪∞ we have σ̂|E ⊃ μ̂ and thus σ̂|k ⊃ μ̂|k ⊃ Γ. One should observe now that F is also a generic splitting field for the splitting off of r hyperbolic planes from φ (instead of φ_{E}). So for this field F instead of the given field K we have a proof of statement (2).

(b) The fields K and F are specialization equivalent over k. We choose a place ρ: K → F ∪ ∞ over k and form the compositum Λ = σ̂ ∘ ρ̂: K → L ∪ ∞. We verify that Λ|k ⊃ Γ and are done.

We have ρ̂ = (ρ, Q(𝔬_{ρ}), ρ̌), σ̂ = (σ, Q(𝔬_{σ}), σ̌), where ρ̌, σ̌ are the characters Q(𝔬_{ρ}) → Q(L), Q(𝔬_{σ}) → Q(L), induced by ρ, σ, i.e., ρ̌(⟨a⟩) = ⟨ρ(a)⟩ for a ∈ 𝔬_{ρ}^{∗}, and σ̌(⟨b⟩) = ⟨σ(b)⟩ for b ∈ 𝔬_{σ}^{∗}. Hence Λ = (σ ∘ ρ, H_{0}, χ) with

H_{0} = {⟨a⟩ ∈ Q(K) | a ∈ 𝔬_{ρ}^{∗}, ⟨ρ(a)⟩ ∈ Q(𝔬_{σ})}

and χ = σ̌ ∘ (ρ̌|H_{0}).

Let Γ = (γ, D, η). For a ∈ k^{∗} with ⟨a⟩ ∈ D we have ⟨a⟩_{K} ∈ Q(𝔬_{ρ}) and ρ̌(⟨a⟩_{K}) = ⟨a⟩_{F}, since ρ is a place over k. Since σ̂|k ⊃ Λ we have ⟨a⟩_{F} ∈ Q(𝔬_{σ}), hence ⟨a⟩_{K} ∈ H_{0} and furthermore

χ(⟨a⟩_{K}) = σ̌(⟨a⟩_{F}) = η(⟨a⟩).

Thus we have verified that Λ|k ⊃ Γ.

As a result of §31 (Theorems 4 and 6) the following corollary of Theorem 1 is obvious.

Scholium. Assume that statements (1) and (2) of the theorem hold. Let φ ⊗ K ≅ ψ ⊥ r × H. Then ψ has SCR with respect to Λ and

Γ_{∗}(φ) ≅ Λ_{∗}(ψ) ⊥ r × H.

Now nothing stands in the way of a generic splitting theory for Γ_{∗}(φ). We note an easy lemma.

Lemma 1. Let Λ: K → L ∪ ∞ be a quadratic place and φ a quadratic form over K which has CR (resp. SCR) with respect to Λ. Further, let j: L ↪ L′

§32 Generic Splitting of Stably Conservative Specialized Quadratic Forms 223

be a field extension, i.e., a trivial place, which is Λ_{∗}(φ)-conservative. Then φ also has CR (resp. SCR) with respect to ĵ ∘ Λ and

(ĵ ∘ Λ)_{∗}(φ) = Λ_{∗}(φ) ⊗_{L} L′.

Proof. Let Λ = (λ, H, χ). We have ĵ = (j, Q(L), Q(j)), where Q(j) denotes the homomorphism from Q(L) to Q(L′), induced by j. We obtain

ĵ ∘ Λ = (j ∘ λ, H, Q(j) ∘ χ),

and of course 𝔬_{j∘λ} = 𝔬_{λ}. This yields the assertion for conservative reduction. The extension to the case of SCR is obvious.

Theorem 2. Let φ be a quadratic form over k and Γ: k → L∪∞ a quadratic place with respect to which φ has SCR. Let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower of φ with higher indices i_{r} and higher kernel forms φ_{r}. Finally, let L′ ⊃ L be a Γ_{∗}(φ)-conservative field extension of L. We choose a quadratic place Λ: K_{m} → L′ ∪ ∞ which extends ĵ ∘ Γ (i.e., Λ|k ⊃ ĵ ∘ Γ), and which in the case m < h cannot be extended to a quadratic place from K_{m+1} to L′. Then

ind(Γ_{∗}(φ) ⊗ L′) = i_{0} + … + i_{m}.

The form φ_{m} has SCR with respect to Λ and

ker(Γ_{∗}(φ) ⊗ L′) = Λ_{∗}(φ_{m}).

Proof. By Lemma 1 we may replace Γ by the quadratic place ĵ ∘ Γ: K → L′ ∪ ∞ and may thus assume without loss of generality that L′ = L. We finish by traversing the proof of §12, Theorem 4 once more and replacing γ by Γ, λ by Λ and FR by SCR everywhere.

If char L ≠ 2 then Theorem 2 is essentially the same theorem as §30, Theorem 4. We have thus found a new proof for that theorem, which is clearly different from the proof in §30. There, a theorem about bad reduction (§30, Theorem 2) was an important tool, for which we have not developed a counterpart which is valid when char L = 2.

We have reached a point where we can envisage applications in the theory of quadratic forms, and in particular in the generic splitting theory, which necessitate the employment of quadratic places, similar to what we did in Chapter III for ordinary places. This would go beyond the scope of this book however. From a scientific point of view we would also be threading on virgin soil. To my knowledge there exist only two publications, [K_{7}] and [KR], in which quadratic places are used. Both works assume characteristic ≠ 2.

224 Chapter IV. Specialization with Respect to Quadratic Places

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Index

AFR (w.r.t. a Q-place) 217 AGR 204 AGR (w.r.t. a Q-place) 217 Arf(φ) 51 Com(φ) 185 CR 218 deg φ 177 det(φ) 9 d(φ) 9 Δ(φ) 99 D(φ) 122 δ(M) 41 φ ≈ ψ 8 φ <_{s} τ 127 φ ∼ ψ 8 φ ⊗ K′ (φ over K, K ⊂ K′) 70 FR 82 FR (w.r.t. a Q-place) 216 GR 61 GR (w.r.t. a Q-place) 203 h(φ) 24, 96 H = ( {0 \atop 1} {1 \atop 0} ) 21 H = [ {0 \atop 1} {1 \atop 0} ] 70 H̃: = ( {0 \atop 1} {1 \atop 0} ) 204 ind(φ) 8 i(ρ_{1}, ρ_{2}) 160 I(k) 180 I^{n}(k) 181 J_{n}(k) 179 j(ρ_{1}, ρ_{2}) 157 ker(φ) 8 k(φ) 23 k(φ) (including for char k = 2) 70 λ̂ 202 λ_{∗}(E) 61 λ_{∗}(φ) 14 Λ_{∗}(φ) 203, 204, 217 λ_{W} (bilinear form) 12 λ_{W} (quadratic form) 56 λ_{W} (more general) 91 Λ_{W} 202 Λ|k 207 M^{h} 45

M^{∧} 41

−M 33

M′ < M (quadratic module) 82 M ∼ N 42 μ_{+}(V) 48 μ(V) 48

N^{∨} 32

N(φ) 122 N^{⊥} 32 ν(φ) 10 𝔬^{h} 45 Pol_{E}(F) 125 Pol_{τ} (φ) 124 ψ < φ (bilinear form) 18 QL(M) 35 ρ(V) 49 SAGR 219 SCR 219 SP(φ) 25 τ|φ 155 W (K) 9 W (K, H) 202 Wq(K, 𝔬) 56 Ŵq(K) 42 Wq(𝔬) 41 W̃q(𝔬) 41 Wqr(𝔬) 41 W̃q(K, 𝔬) 56

affine standard tower 100

almost fair reduction (w.r.t. a Q-place)

217 almost good reduction (w.r.t. a

Q-place) 204, 217 anisotropic (bilinear form) 7 anisotropic (quadratic module) 38 Arf invariant 51 associated quadratic space 41 base extension 31 bilinear form

– anisotropic 7

– good reduction 12

– hyperbolic 7

– kernel form 8

– λ-unimodular 12

– metabolic 7

– nondegenerate 12

– representation 18

– Substitution Principle 18

– Witt class 8

– Witt decomposition 7

– Witt equivalence 8

– (Witt) index 8 bilinear linkage number 160 bilinear 𝔬-module 13 bilinear Pfister form 121 bilinear space 15

– good reduction 57

– λ-modular decomposition 57

– Lagrangian subspace 16

– metabolic 16

– nondegenerate 15

– 𝔬-modular decomposition 57

– subspace 15

– totally isotropic subspace 16 bilinearly r-linked Pfister forms 159

close Pfister neighbour 127 companion 185 complementary form 164 compositum of Q-places 221 conservative reduction 218

defect 41 defective quadratic Witt set 42 definable over a field 174 definable over a field (for leading forms)

193 definable over a field by a Pfister form

(for leading forms) 194 degree 177 determinant 9 dimension index 10 discriminant 99 discriminant algebra 99 disobedient quadratic space 50 divisibility 155

equivalent fields 21 excellent form 165 expansion 207 extension 208

fair reduction 82

fair reduction (w.r.t. a Q-place) 216 φ-conservative field extension 93 free hyperbolic module 32 free quadratic module 29 fundamental ideal 180

generic separable zero field 78 generic splitting field of level r 96 generic splitting tower 24, 79, 94 generic zero field 22, 70, 96 good reduction 61 good reduction (bilinear form) 12 good reduction (bilinear space) 57 good reduction (quadratic space) 47 good reduction (w.r.t. a Q-place) 203

height 24, 96

higher complementary form 165 higher index 24, 94 higher kernel form 24, 94 hyperbolic (bilinear form) 7 hyperbolic pair 39 hyperbolic quadratic module (over a

valuation domain) 43

inessential field extension 22

inner type 99

isotropic (quadratic module) 38

kernel form (bilinear form) 8

kernel module 42

kernel module (quadratic module) 40

λ-modular decomposition 64 λ-modular decomposition (bilinear

space) 57 λ-modular decomposition (quadratic

space) 53 λ-unimodular (bilinear form) 12 Λ-modular decomposition 204, 217 Λ-unimodular 203, 216 Lagrangian subspace (bilinear space)

16 leading field 178 leading form 178

metabolic (bilinear form) 7 metabolic (bilinear space) 16

neighbouring Pfister form 164 nondegenerate (bilinear form) 12 nondegenerate (bilinear space) 15 nondegenerate (quadratic module) 35 norm group 122 Norm Theorem 142

𝔬-modular decomposition (bilinear

space) 57 𝔬-modular decomposition (quadratic

space) 53 obedient quadratic space 50 outer type 99

partial generic splitting field 96 Pfister form 26

– bilinear 121

– quadratic 121 Pfister neighbour 163 polar 124 primitive vector 35 projective standard tower 100 pure part of a Pfister form 26, 122

Q-place 202 (Q(L)-valued) character 202 quadratic form 29

– Substitution Principle 89 quadratic linkage number 157 quadratic module 29

– anisotropic 38

– isotropic 38

– kernel module 40

– nondegenerate 35

– reduced nondegenerate 82

– regular 36

– representation 82

– strictly regular 36

– weakly λ-modular decomposition 90

– weakly 𝔬-modular decomposition 90

– weakly obedient 90

– Witt class 40

– Witt decomposition 40

– Witt equivalence 40

– Witt index 40 quadratic Pfister form 121 quadratic place 202 quadratic space 41, 47

– disobedient 50

– good reduction 47

– λ-modular decomposition 53

– 𝔬-modular decomposition 53 – obedient 50 quadratic space over 𝔬 61 quadratically henselian 43 quadratically r-linked Pfister forms

157 quasilinear 35 quasilinear part 35

r-linked Pfister forms 157 r-th complementary form 165 radical 35 realization of a Q-place 211 reduced nondegenerate quadratic

module 82 reduction of an anisotropic quadratic

module 49 regular (quadratic module) 36 regular field extension 101 regular generic splitting tower 102 representation (bilinear form) 18 representation (quadratic module) 82 represented element 122 restriction of a Q-place 207 round form 151

separable field extension 77

signed determinant 9 similarity norm 122 specialization 14 specialization (in the presence of FR)

86 specialization (w.r.t. a Q-place) 203,

204, 217 specialization equivalent fields 21 splitting pattern 25 stable isometry 8 stably almost good reduction 219 stably conservative reduction 219 strict extension 208 strict realization of a Q-place 211 strictly regular (quadratic module) 36 strictly regular hull 127 strongly multiplicative form 147 Subform Theorem 131 subspace (bilinear space) 15 Substitution Principle (bilinear form)

18 Substitution Principle (quadratic form)

89

tensor product 31

total generic splitting field 96 totally isotropic subspace (bilinear

space) 16

weak specialization 55 weak specialization (more general) 91 weakly λ-modular decomposition

(quadratic module) 90 weakly Λ-modular decomposition 217 weakly Λ-unimodular 216 weakly 𝔬-modular decomposition

(quadratic module) 90 weakly obedient quadratic module 90 Witt class (bilinear form) 8 Witt class (quadratic module) 40 Witt decomposition (bilinear form) 7 Witt decomposition (quadratic module)

40 Witt equivalence (bilinear form) 8 Witt equivalence (over a field) 42 Witt equivalence (quadratic module)

40 (Witt) index (bilinear form) 8 Witt index (quadratic module) 40 Witt ring 9