Generic Splitting Towers and Generic Splitting Preparation of Quadratic Forms

Manfred Knebusch and Ulf Rehmann

Abstract. This manuscript describes how a generic splitting tower of a regular anisotropic quadratic form digests the form down to a form which is totally split.

Introduction

We work with quadratic forms on finite dimensional vector spaces over an arbitrary field k. We call such a form q: V → k regular if the radical V^{⊥} of the associated bilinear form B_{q} has dimension ≤ 1 and the quasilinear part q|V^{⊥} of q is anisotropic. If k has characteristic char k ≠ 2 this means that V^{⊥} = {0}. If char k = 2 it means that either V^{⊥} = {0} or V^{⊥} = kv with q(v) ≠ 0. In the present article a “form” always means a regular quadratic form. Our first goal is to develop a generic splitting theory of forms. Such a theory has been given in [K_{2}] for the case of char k ≠ 2. Without any restriction on the characteristic, a generic splitting theory for complete quotients of reductive groups was given in [KR], which is closely related to our topic. In §1 we present a generic splitting theory of forms in a somewhat different manner than in [K_{2}]. We start with a key result from [KR] (cf. Theorem 1.3 below), then develop the notion of a generic splitting tower of a given form q over k with associated higher indices and kernel forms, and finally explain how such a tower (K_{r} | 0 ≤ r ≤ h) together with the sequence of higher kernel forms (q_{r} | 0 ≤ r ≤ h) of q controls the splitting of q⊗L into a sum of hyperbolic planes and an anisotropic form (called the anisotropic part or kernel form of q ⊗ L), cf. 1.19 below. More generally we explain how the generic splitting tower (K_{r} | 0 ≤ r ≤ h) together with (q_{r} | 0 ≤ r ≤ h) controls the splitting of the specialization γ_{∗}(q) of q by a place γ: k → L ∪ ∞, if q has good reduction under q, cf. Theorem 1.18 below. Then in §2 we study how a generic splitting tower of q ⊗ L can be constructed from a generic splitting tower of q for any field extension L/k. All these results are an expansion of the corresponding results in [K_{2}] to fields of any characteristic. We mention that a reasonable generic splitting theory holds more generally for a quadratic form q: V → k such that the quasilinear part q|V^{⊥} is anisotropic, without the additional assumption dim V^{⊥} ≤ 1. This needs more work. It will be contained in the forthcoming book [K_{5}]. In §3 we prove that for any form q over k there exists a generic splitting tower (K_{r} | 0 ≤ r ≤ h) of q which contains a subtower (K′_{r} | 0 ≤ r ≤ h) of field extensions of k such that K′_{r}/K′_{r−1} is purely transcendental, and such that the anisotropic part of q ⊗K_{r} can be defined over K′_{r} for every r ∈ [1, h]. {We have K′_{0} = K_{0} = k.} This result, which may be surprising at first glance, leads us in §4 to the second theme of this article, namely generic splitting preparations (Def. 4.3) and the closely related generic splitting decompositions (Def. 4.8) of a form q. We focus now on the second notion, since its meaning can slightly more easily be grasped than that of the first (more general) notion. A generic splitting decomposition of a form q over k consists of a purely transcendental field extension K′/k and an orthogonal decomposition

1991 Mathematics Subject Classification. Primary 11E04, 11E81.

Key words and phrases. quadratic forms, generic splitting.

This research was supported by the European Union TMR Network FMRX CT-97-0107 ”Algebraic K-Theory, Linear Algebraic Groups and Related Structures”, and by the Sonderforschungsbereich ”Diskrete Mathematik” at the University of Bielefeld.

q ⊗ K′ ≅ η_{0} ⊥ η_{1} ⊥ ⋯ ⊥ η_{h} ⊥ φ_{h}  (∗)

with certain properties. In particular, dim φ_{h} ≤ 1, all η_{i} have even dimension, and η_{0} is the hyperbolic part of q ⊗ K′ (which comes from the hyperbolic part of q by going up from k to K′). The generic splitting decomposition in a certain sense controls the splitting behavior of q ⊗ L for any field extension L of k, more generally of γ_{∗}(q), for any place γ: k → L ∪ ∞ such that q has good reduction under γ. This control can be made explicit in much the same way as the control by generic splitting towers, using “quadratic places” (or “Q-places” for short) instead of ordinary places, cf. §6. Quadratic places have been introduced in the recent article [K_{4}] and used there for another purpose. We recapitulate here what is necessary in §5. We are sorry to say that our theory in §6 demands that the occurring fields have characteristic ≠ 2. This is forced by the article [K_{4}], where the specialization theory of forms under quadratic places is only done in the case of characteristics ≠ 2. It seems that major new work and probably also new concepts are needed to establish a specialization theory of forms under quadratic places in all characteristics. An overall idea behind generic splitting decompositions is the following. If we allow for the form q over k a suitable linear change of coordinates with coefficients in a purely transcendental field extension K′ ⊃ k, then the form – now called q ⊗ K′ – decomposes orthogonally into subforms η_{0}, η_{1}, …, η_{h}, φ_{h} such that the forms η_{k} ⊥ ⋯ ⊥ η_{h} ⊥ φ_{h} with 1 ≤ k ≤ h give the higher kernel forms of q, when we go up further from K′ to suitable field extensions of K′. Thus, after the change of coordinates, the form q is “well prepared” for an investigation of its splitting behavior. This reminds a little of the Weierstrass preparation theorem, where an analytic function germ becomes well prepared after a linear change of coordinates. In contrast to Weierstrass preparation we allow a purely transcendental field extension for the coefficients of the linear change of coordinates. But no essential information about the form q is lost by passing from q to q ⊗ K′, since q is the specialization λ_{∗}(q ⊗ K′) of q ⊗ K′ under any place λ: K′ → k ∪ ∞ over k.

The idea behind generic splitting preparations is similar. {Generic splitting decompositions form a special class of generic splitting preparations.} Now the forms η_{i} are defined over fields K′_{i} such that K′_{0} = k and every K′_{i} is a purely transcendental extension of K′_{i−1}. Generic splitting decompositions and, more generally, generic splitting preparations give new possibilities for manipulations with forms. For example, if q ⊗ K′ ≅ η_{0} ⊥ ⋯ ⊥ η_{h} ⊥ φ_{h} is a generic splitting decomposition of q, then we may look how many hyperbolic planes split off in q ⊗ E_{r} for E_{r} the total generic splitting field of one of the summands η_{r}. We do not enter these matters here, leaving all experiments to the future and to the interested reader.

§1. Generic splitting in all characteristics

1.0. Notations. For a, b elements of a field k we denote the form aξ^{2} + ξη + bη^{2} over k by [a, b]. Since we only allow regular forms, we demand 1 − 4ab ≠ 0. By H: = [0, 0] we denote the hyperbolic plane. If q is a (regular quadratic) form over k then we have the Witt decomposition ([W], [A]) q ≅ r × H ⊥ φ with an anisotropic form φ and r ∈ ℕ_{0}. We call r the index of q and write r = ind (q). We further call φ the kernel form ^{1)} or anisotropic part of q and use both notations φ = ker(q), φ = q_{an}. If L ⊃ k is a field extension then q ⊗ L or q_{L} denotes the form over L obtained from q by extension of the base field k to L. Thus, if q lives on the k-vector space V, then q ⊗ L lives on the L-vector space L ⊗_{k} V. A major theme of this article is the study of ind (q ⊗ L) and ker(q ⊗ L) for varying extensions L/k. dim q denotes the dimension of the vector space V on which q lives, i.e., the number of variables occurring in the form q. We have dim q = dim(q ⊗ L). The zero form q = 0 is not excluded. Then V = {0} and dim q = 0. We say that q splits totally if dim(q_{an}) ≤ 1. This is equivalent to ind (q) = [dim q/2]. For another form φ over k we write φ < q if φ is isometric to a subform of q (including the case φ ≅ q).

1.1. Definition/Further notations. If q ≠ 0 and dim q is even, let

δq: = {(discriminant of q) ∈ k^{∗}/k^{∗2} if char k ≠ 2 (Arf invariant of q) ∈ k^{+}/℘k if char k = 2,

p_{δq}(X): = {X^{2} − δq  if char k ≠ 2 X^{2} + X + δq  if char k = 2.

The separable polynomial p_{δq}(X) splits over k if and only if δq is trivial. If dim q is odd or if δq is trivial we say that q is of inner type, otherwise we say that q is of outer type. These notions are adjusted to the corresponding notions in the theory of reductive groups. q is inner (resp. outer) if and only if SO(q) is inner (resp. outer). {N.B. SO(q) is almost simple for dim q ≥ 3 since the form q is regular.} We define

k_{δq}: = {k if q is of inner type k[X]/p_{δq}(X) if q is of outer type.

For i = 1, …, [dim q/2], we denote by V_{i}(q) the projective variety of totally isotropic subspaces of dimension i in the underlying space of q, and by k_{i}(q) we denote the

1) = “Kernform” in [W] function field of V_{i}(q), unless dim q = 2. In the latter case V_{1} consists of two irreducible components defined over k_{δq}. We then set k_{1}(q) = k_{δq}. In general, we will also write k(q) = k_{1}(q), which is, with the above interpretation, the function field of the quadric V_{1}(q) associated to q by the equation q = 0. □

1.2. Lemma. Let q be a (regular quadratic) form over k.

i) If K/k is any field extension such that q_{K} is of inner type, then K contains a subfield isomorphic to k_{δq}.

ii) Let i ∈ {1, …, [dim q/2]}. If q is of inner type or if i ≤ dim q/2 − 1, then V_{i}(q) is defined over k. If q is of outer type, then V_{dim q/2}(q) is defined over k_{δq}.

iii) V_{i}(q) is geometrically irreducible unless q is of outer type and i = dim q/2−1, in which case it decomposes, over k_{δq}, into two geometrically irreducible components isomorphic to V_{dim q/2}(q).

Proof. i): Let dim q be even. Clearly q_{K} is inner if and only if the polynomial p_{δq} has a zero in K, hence i) follows. ii), iii): Since the stabilizer of an i-dimensional totally isotropic subspace of the underlying space of q is a parabolic subgroup of SO(q), the statements follow from [KR, 3.7, p. 44f]. □

A key observation for the generic splitting theory of quadratic forms is the following theorem, which has a generalization for arbitrary homogeneous projective varieties [KR, 3.16, p. 47]:

1.3. Theorem. Let q be a form over k, let F_{i} denote the function field of V_{i}(q) as a regular extension of k resp. k_{δq} according to 1.2.ii, and let L/k be a field extension. The following statements are equivalent.

i) ind (q ⊗ L) ≥ i.

ii) The projective variety V_{i}(q) has an L-rational point.

iii) There is a k-place F_{i} → L ∪ ∞.

iv) L contains a subfield isomorphic to the algebraic closure F_{i}^{0} of k in F_{i}, and the free composite LF_{i} over F_{i}^{0} is a purely transcendental extension of L.

Remark. Only in the case of an outer q and i = dim q/2, we have F_{i}^{0} = k_{δq} ≠ k; in all other cases, i.e., if q is inner or i ≤ dim q/2 − 1, we have F_{i}^{0} = k in iv).

Proof of 1.3. The equivalence of i) and ii) is obvious. The other equivalences follow from [KR, 3.16, p.47], again after observing that the stabilizer of an idimensional totally isotropic subspace is a parabolic subgroup of SO(q). □

1.4. Definition. We call two field extensions K ⊃ k and L ⊃ k specialization equivalent over k, and we write K ∼_{k} L, if there exists a place from K to L over k and also a place from L to K over k. □

1.5. Corollary. If q′ = l × H ⊥ q, and if F′_{i} is the function field of V_{i}(q′), then F′_{l+i} and F_{i} are specialization equivalent over k. ^{2)} .

2) We will denote the hyperbolic plane [0, 0] over any field by H Proof. This is obvious by the equivalence of i) and iii) in 1.3. □

In the following q is a (regular quadratic) form over k. We want to associate to q partial generic splitting fields and partial generic splitting towers as has been done in [K_{2}] for char k ≠ 2. We will proceed in a different way than in [K_{2}], starting with a formal consequence of Theorem 1.3.

1.6. Corollary. Let L and L′ be field extensions of k. Assume there exists a place λ: L → L′ ∪ ∞ over k. Then ind (q ⊗ L′) ≥ ind (q ⊗ L).

Proof. Let i: = ind (q⊗L). By the theorem there exists a place ρ: F_{i} → L∪∞ over k. Then λ∘ρ is a place from F_{i} to L′ ∪∞. Again by the theorem, ind (q ⊗L′) ≥ i.□

1.7. Definition. (Cf. [HR_{1}]). The splitting pattern SP(q) is the (naturally ordered) sequence of Witt indices ind (q ⊗ L) with L running through all field extensions of k. □

Notice that the sequence SP(q) is finite, consisting of at most [dim q/2]+1 elements j_{0} < j_{1} < ⋯ < j_{h}. Of course, j_{0} = ind (q) and j_{h} = [dim q/2]. We call h the height of q, and we write h = h(q). Notice also that SP(q) is the sequence of all numbers i ≤ [dim q/2] with ind (q ⊗ F_{i}) = i.

1.8. Definition. Let r ∈ {0, 1, …, h} = [0, h]. A generic splitting field of q of level r is a field extension F/k with the following properties:

a) ind (q ⊗ F) = j_{r}.

b) For every field L ⊃ k with ind (q⊗L) ≥ j_{r} there exists a place λ: F → L∪∞

over k. Such a field extension F/k, for any level r, is also called a partial generic splitting field of q, and, in the case r = h, a total generic splitting field of k. □

It is evident from the definitions and from Corollary 1.6 that, if K is a generic splitting field of q of some level r and L is a field extension of k, then L is a generic splitting field of q of level r if and only if K and L are specialization equivalent over k.

1.9. Proposition. Let r ∈ [0, h]. All the fields F_{i} from Theorem 1.3 with j_{r−1} < i ≤ j_{r} are generic splitting fields of q of level r. {Read j_{−1} = −1.} In particular, the fields k(q_{an}), F_{j_{0}+1}, …, F_{j_{1}} are generic splitting fields of q of level 1.

Proof. By Theorem 1.3, we certainly have ind (q⊗F_{i}) ≥ i, hence ind (q⊗F_{i}) ≥ j_{r}. If L/k is any field extension with ind (q ⊗ L) ≥ j_{r}, then, again by Theorem 1.3, there exists a place λ: F_{i} → L ∪ ∞ over k. Thus condition b) in Definition 1.8 is fulfilled. We can choose L as an extension of k with ind (q ⊗ L) = j_{r}. By Corollary 1.6 we have ind (q ⊗ F_{i}) ≤ ind (q ⊗ L). Thus ind (q ⊗ F_{i}) = j_{r}. Moreover, since k(q_{an}) is the function field of V_{1}(q_{an}), it follows from 1.5 that this field is specialization equivalent over k to F_{j_{0}+1}. □

1.10. Corollary. If F is a generic splitting field of q (of some level r), then the algebraic closure of k in F is always k, except if q is outer and ind (q_{F}) = dim q/2, in which case it is k_{δq}.

Proof. Clearly F ∼_{k} F_{i} for i = ind q_{F}, hence we have k-places from F to F_{i} and vice versa, which are of course injective on the algebraic closure of k in F resp. F_{i}. Our claim now follows from 1.3 and the remark after 1.3. □

1.11. Scholium. Let (K_{r} | 0 ≤ r ≤ h) be a sequence of field extensions of k such that for each r ∈ [0, h] the field K_{r} is a generic splitting field of q of level r. Let L/k be a field extension of k. We choose s ∈ [0, h] maximal such that there exists a place from K_{s} to L over k. Then ind (q ⊗ L) = j_{s}.

Proof. By Corollary 1.6 we have ind (q ⊗ L) ≥ j_{s}. Suppose that ind (q ⊗ L) > j_{s}. Then ind (q ⊗ L) = j_{r} for some r ∈ [0, h] with r > s. Thus there exists a place from K_{r} to L over k. This contradicts the maximality of s. We conclude that ind (q ⊗ L) = j_{s}. □

If q is anisotropic and dim q ≥ 2, then a generic splitting field of q of level 1 is called a generic zero field of q. Proposition 1.9 tells us that, in general, F_{j_{0}+1} and k(q_{an}) are generic zero fields of q_{an}. {N.B. The notion of generic zero field has also been established if q is isotropic, cf. [K_{2}, p. 69]. Then it still is true that k(q) is a generic zero field of q.}

1.12. Definition. A generic splitting tower of q is a sequence of field extensions K_{0} ⊂ ⋯ ⊂ K_{h} of k such that K_{0} is specialization equivalent over k with k, and such that K_{r+1} is specialization equivalent over K_{r} with K_{r}(q_{K_{r},an}). ^{3)} In particular, the inductively defined sequence K_{0} = k, K_{r+1} = K_{r}(q_{K_{r},an}) is the standard generic splitting tower of q (cf. [K_{2}, p. 78]). We call q_{r}: = (q_{K_{r}})_{an} the r-th higher kernel form of q (with respect to the tower). We define i_{0}: = ind q and i_{r}: = ind q_{r−1} ⊗ K_{r} for 1 ≤ r ≤ h, and we call i_{r} the r-th higher index of q (0 ≤ r ≤ h).

1.13. Theorem. If K_{0} ⊂ ⋯ ⊂ K_{h} is a generic splitting tower of q, then, for every r ∈ [0, h], the field K_{r} is a generic splitting field of q of level r.

Proof. We denote the function fields of the varieties V_{i}(q) by F_{i}, as in 1.3. By 1.9, it suffices to show K_{r} ∼_{k} F_{j_{r}}, for every r ≥ 0. For r = 0 this is obvious, since F_{j_{0}} is a purely transcendental extension of k. For r = 1 we have, by 1.5, applied to q_{K_{0}} = j_{0} × H ⊥ q_{K_{0},an},

K_{1} ∼_{K_{0}} K_{0}(q_{K_{0},an}) ∼_{K_{0}} F_{j_{0}+1}K_{0} ∼_{k} F_{j_{0}+1},

hence K_{1} ∼_{k} F_{j_{0}+1} ∼_{k} F_{j_{1}}, by 1.9. We proceed by induction on dim q_{an}. By induction assumption, our claim is true for q_{1} := q_{K_{1},an} over K_{1}, and hence for q_{K_{1}} = (j_{0} + j_{1}) × H ⊥ q_{1} by 1.5. That is, for r ≥ 1, the field K_{r} is a generic splitting field of q_{K_{1}} of level r − 1, and, as such, specialization equivalent over K_{1} with the function field of V_{j_{r}}(q_{K_{1}}) ≅ V_{j_{r}}(q)×_{k} K_{1} resp. ≅ V_{j_{r}}(q) ×_{k_{δq}} K_{1}k_{δq}, which is F_{j_{r}} · K_{1} (free product over k resp. k_{δq}). Hence it remains to show that F_{j_{r}} · K_{1} is specialization equivalent to F_{j_{r}} over k. We have a trivial k-place F_{j_{r}} → F_{j_{r}} · K_{1} ∪ ∞. On the other hand, since r ≥ 1, we also have a k-place K_{1} → F_{j_{r}} ∪ ∞, which gives us a k-place F_{j_{r}} · K_{1} → F_{j_{r}} ∪ ∞. □

The rest of this paragraph will be used in paragraphs 5 and 6 only. For the next statements we need the notion of “good reduction” of a quadratic form.

3) This definition of generic splitting towers is slightly broader than the definition in [K_{2}, p.78]. There it is demanded that K_{0} = k.

1.14. Definition/Remark. Let q: K^{n} → K be a quadratic form over a field K and λ: K → L ∪ ∞ be a place to a second field L. Let 𝔬 = 𝔬_{λ} denote the valuation ring of λ.

a) We say that q has good reduction (abbreviated: GR) under the place λ, if there exists a linear change of coordinates T ∈ GL(n, K) such that (x: = (x_{1}, …, x_{n}))

q(T x) = ∑_{i≤j} a_{ij}x_{i}x_{j}

with coefficients a_{ij} ∈ 𝔬, and such that the form ∑_{i≤j} λ(a_{ij})x_{i}x_{j} over L is

regular.

b) In this situation it can be proved that, up to isometry, the form

∑_{i≤j} λ(a_{ij})x_{i}x_{j}

does not depend on the choice of T (cf. [K_{1}, Lemma 2.8], [K_{5}, §8]). Abusively we denote this form by λ_{∗}(q), and we call λ_{∗}(q) “the” specialization of q under λ.

c) Let q be a regular form over k, let K, L be field extensions of k and let λ : K → L ∪ ∞ be a k-place with valuation ring 𝔬. Then q_{K} has GR under λ and λ_{∗}(q_{K}) = q_{L}. By Lemma 1.14.b below it follows that also q_{K,an} has GR under λ. □

If q has GR under λ then certainly q itself is regular. Moreover it can be proved that

(∗)  q ≅ [a_{1}, b_{1}] ⊥ ⋯ ⊥ [a_{m}, b_{m}] ( ⊥ [ε] )

with elements a_{i}, b_{i} ∈ 𝔬 and ε ∈ 𝔬^{∗} (cf. [K_{1}], [K_{5}, §6]). Here the last summand [ε] denotes the form εX^{2} in one variable X. It appears if and only if n = dim q is odd. Of course, (∗) implies

λ_{∗}(q) ≅ [λ(a_{1}), λ(b_{1})] ⊥ ⋯ ⊥ [λ(a_{m}), λ(b_{m})] ( ⊥ [λ(ε)] ).

1.15. Lemma. Let q and q′ be forms over K, and assume that dim q is even. a) If q and q′ have GR under λ then q ⊥ q′ has GR under λ, and

λ_{∗}(q ⊥ q′) ≅ λ_{∗}(q) ⊥ λ_{∗}(q′).

b) If q and q ⊥ q′ have GR under λ, then q′ has GR under λ.

Proof. Part a) of this lemma is trivial, but b) needs a proof. A proof can be found in [K_{1}, §2] in the case that also q′ has even dimension, and in [K_{5}, §8] in general. □

Part b) will be crucial for the arguments to follow.

1.16. Proposition. Let λ: K → L ∪ ∞ be a place and φ a form over K with GR under λ. Then φ_{0}: = ker(φ) has again GR under λ and λ_{∗}(φ) ∼ λ_{∗}(φ_{0}), ind (λ_{∗}(φ)) ≥ ind (φ). If ind (λ_{∗}(φ)) = ind (φ), then ker λ_{∗}(φ) = λ_{∗}(φ_{0}).

Proof. Let φ_{0}: = ker φ. We have φ ≅ j × H ⊥ φ_{0} with j = ind (φ). The form j × H has GR under λ. By Lemma 1.15 it follows that φ_{0} has GR under λ and λ_{∗}(φ) ≅ j × H ⊥ λ_{∗}(φ_{0}). Now all the claims are evident. □

1.17. Proposition. Let φ be an anisotropic form over K of dimension ≥ 2 which has GR under a place λ: K → L ∪ ∞. Let K_{1} ⊃ K be a generic zero field of φ. Then λ_{∗}(φ) is isotropic if and only if λ extends to a place μ: K_{1} → L ∪ ∞.

Sketch of Proof. a) If λ extends to a place μ: K_{1} → L ∪ ∞ then it is obvious that φ ⊗ K_{1} has GR under μ and μ_{∗}(φ ⊗ K_{1}) = λ_{∗}(φ). Since φ ⊗ K_{1} is isotropic we conclude by Proposition 1.16 that λ_{∗}(φ) is isotropic.

b) Assume now that λ_{∗}(φ) is isotropic. We denote this form by \overline{φ} for short. By use of elementary valuation theory it is rather easy to extend λ to a place λ̃: K(φ) → L(\overline{φ})∪∞ (cf. [K_{5}, §9]; here we do not need that \overline{φ} is isotropic). Since \overline{φ} is isotropic the field extension L(\overline{φ})/L is purely transcendental (cf. Th. 1.3). Thus there exists a place ρ: L(\overline{φ}) → L ∪ ∞ over L. Now ρ ∘ λ̃: K(φ) → L ∪ ∞ is a place extending λ. Since K(φ) ∼_{K} K_{1}, there exists also a place μ: K_{1} → L ∪ ∞ extending λ. □

We return to our form q over k.

1.18. Theorem. Let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower of q with higher indices i_{r} and higher kernel forms q_{r} (0 ≤ r ≤ h). Let γ: k → L ∪ ∞ be a place such that q has GR under γ. Moreover let m ∈ [0, h] and λ: K_{m} → L ∪ ∞ be a place extending γ. Assume in the case m < h that λ does not extend to a place from K_{m+1} to L. Then ind (γ_{∗}(q)) = i_{0} + ⋯ + i_{m} = j_{m}. The form q_{m} has GR under λ and ker(γ_{∗}(q)) ≅ λ_{∗}(q_{m}).

Proof. We have i_{0} + ⋯ + i_{m} = j_{m} and q ⊗ K_{m} ≅ j_{m} × H ⊥ q_{m}. This implies, that q_{m} has GR under λ and

γ_{∗}(q) = λ_{∗}(q ⊗ K_{m}) ≅ j_{m}×H ⊥ λ_{∗}(q_{m})

(cf. Proof of Prop. 1.16.) It remains to prove that λ_{∗}(q_{m}) is anisotropic. This is trivial if m = h. Assume now that m < h. If λ_{∗}(q_{m}) would be isotropic then Proposition 1.17 would imply that λ extends to a place from K_{m+1} to L, contradicting our assumptions in the theorem. Thus λ_{∗}(q_{m}) is anisotropic. □

Applying the theorem to the special case that L is a field extension of k and γ is the trivial place k ↪ L, we obtain a result on the Witt decomposition of q ⊗ L which is much stronger than 1.11.

1.19. Corollary. Let (K_{r} | 0 ≤ r ≤ h) be a generic splitting tower of q. If L/k is a field extension and ind (q ⊗L) = j_{m}, and if ρ: K_{r} → L∪∞ is a place over k for some r ∈ [0, h], then r ≤ m and ρ extends to a place λ: K_{m} → L∪∞. For every such place λ the kernel form q_{m} of q ⊗ K_{m} has GR under λ and λ_{∗}(q_{m}) ≅ ker(q ⊗ L).□

An easy consequence is the following statement.

1.20. Scholium. (“Uniqueness” of generic splitting towers and higher kernel forms). Let (K_{r} | 0 ≤ r ≤ h) and (K′_{r} | 0 ≤ r ≤ h) be generic splitting towers of q with associated sequences of higher indices (i_{r} | 0 ≤ r ≤ h), (i′_{r} | 0 ≤ r ≤ h) and sequences of kernel forms (q_{r} | 0 ≤ r ≤ h), (q′_{r} | 0 ≤ r ≤ h). Then i_{r} = i′_{r} for every r ∈ [0, h]. There exists a place λ: K_{h} → K′_{h} ∪ ∞ over k which restricts to a place λ_{r}: K_{r} → K′_{r} ∪∞ for every r ∈ [0, h]. If there is given a place μ: K_{r} → K′_{r} ∪∞ over k for some r ∈ [0, h] then q_{r} has good reduction under μ and μ_{∗}(q_{r}) ≅ q′_{r}. □

§2. Behavior of generic splitting fields and generic splitting towers under base field extension

2.1. Definition/Remark. If K/k is a partial generic splitting field of q of some level r, then we denote the algebraic closure of k in K by K^{∘}. The extension K^{∘}/k is k or k_{δq} (cf. 1.10).

For systematic reasons we retain the notation K^{∘} for later use, although most often K^{∘} = k.

2.2. Definition. We call a generic splitting field K of q of some level r ∈ [0, h] regular, if K is regular over the algebraic closure K^{∘} of k in K. We then denote by L · K, or more precisely by L ·_{k} K, the free composite of L · K^{∘} and K over K^{∘}. Explanation. Here we have to read K^{∘} = k, L · K^{∘} = L if r < h or if r = h and q is inner. If r = h and q is outer we have two cases. Either L splits the discriminant of q. In this case K^{∘} = k_{δq} embeds into L and we read L · K^{∘} = L. Or L does not split δq. In this case L · K^{∘} = L ⊗_{k} K^{∘} = L_{δ(q⊗L)}. □

2.3. Definition. We call a generic splitting tower (K_{r} | 0 ≤ r ≤ h) of q regular if K_{r}/K_{r−1} is a regular field extension for every r with 1 ≤ r < h, and also for r = h, if the form q is inner. If r = h and q is outer, we demand that K_{h} is regular over the composite K_{h−1} · K_{h}^{∘} = K_{h−1} · k_{δq} = K_{h−1} ⊗_{k} k_{δq} over k. □

Let L/k be any field extension. We want to construct partial generic splitting fields and generic splitting towers for q ⊗ L from corresponding data for q. Assume that (K_{r} | 0 ≤ r ≤ h) is a regular generic splitting tower of q. For every r ∈ [0, h] we have the free composite L · K_{r} = L ·_{k} K_{r} as explained in 2.2. (The existence of the free products L·K_{r} is the only assumption needed for the following theorem. This is generally true if either L or K_{r} is regular over k resp. K^{0}. Thus, instead of the regularity of the generic splitting tower, we could also assume that the field L is separable over k.)

2.4. Theorem. Let J = (r_{0}, … r_{e}) denote the sequence of increasing numbers r ∈ {0, …, h} such that ind (q ⊗ K_{r}) = ind (q ⊗ L · K_{r}).

a) Then the sequence

L · K_{r_{0}} ⊂ L · K_{r_{1}} ⊂ ⋯ ⊂ L · K_{r_{e}}

is a regular generic splitting tower of q ⊗ L·K_{0}, and hence of q ⊗ L.

b) For every r ∈ [0, h] \ J we have an L · K_{r}-place L · K_{r+1} → L · K_{r} ∪ ∞.

Proof. The claim is obvious if dim q_{an} ≤ 1. We proceed by induction on dim q_{an}. Let r′ := min J \ {0}. The induction hypothesis, applied to q_{K_{1}}, gives a regular generic splitting tower L · K_{r′} ⊂ ⋯ ⊂ L · K_{r_{e}} for q_{L·K_{1}}, as well as b) for r ≥ 1. In particular, the latter implies that L · K_{r′} ∼_{L·K_{1}} L · K_{1} ∼_{L·K_{0}} L · K_{0}(q_{L·K_{0},an}), and this proves a). It remains to show b) for r = 0. But 0 ∉ J means ind q_{L} > ind q, hence ind q_{L·K_{0}} > ind q_{K_{0}}. Therefore there is a K_{0}-place K_{1} → L·K_{0}∪∞, which yields an L·K_{0}-place L · K_{1} → L · K_{0} ∪ ∞. □

In [K_{2}, p. 85] another proof of Theorem 2.4 and its corollary has been given, which clearly remains valid if char k = 2. We believe that the present proof albeit shorter gives more insight than the proof in [K_{2}].

The sequence SP(q ⊗ L) is a subsequence of SP(q) = (j_{0}, …, j_{h}), say SP(q ⊗ L) = (j_{t(0)}, …, j_{t(e)}), with

0 ≤ t(0) < t(1) < ⋯ < t(e) = h.

{ It is evident, that j_{t(e)} = j_{h} = [dim q/2] ∈ SP(q ⊗ L.}

It follows from Theorem 2.4 that the t(i) coincide with the numbers r_{i} there. Thus we have the following corollary.

2.5. Corollary. a) For every s ∈ [0, e] the anisotropic part of q ⊗ L · K_{t(s)} is q_{t(s)} ⊗ L · K_{t(s)}.

b) SP(q ⊗ L) is the sequence of all r ∈ [0, h] such that the form q_{r} ⊗ L · K_{r} is anisotropic. □

2.6. Proposition. Let K be a regular generic splitting field of q of some level r ∈ [0, h]. Then L · K is a generic splitting field of q ⊗ L of level s, where s ∈ [0, e] is the number with t(s − 1) < r ≤ t(s). {Read t(−1) = −1.}

Proof. We return to the fields F_{i} in Theorem 1.3. Let i: = j_{r}. We have K ∼_{k} F_{i} by Proposition 1.9. This implies K · L ∼_{L} F_{i} · L. Thus it suffices to prove the claim for F_{i} instead of K. Now L · F_{i} is the function field of the variety V_{i}(q ⊗ L). Proposition 1.9 gives the claim. □

For later use, it is convenient to insert a digression about “inessential” field extensions.

2.7. Definition. We call a field extension E/k inessential, if there exists a place α: E → k ∪ ∞ over k, i.e., E ∼_{k} k. □

The idea behind this definition is that, if E/k is inessential, then q⊗E has essentially the same splitting behavior as q. This will now be verified. We know already from Corollary 1.6 that ind (q ⊗ E) = ind (q), hence ker(q ⊗ E) = ker(q) ⊗ E.

2.8. Corollary. Assume again that E/k is an inessential field extension.

i) If (E_{r} | 0 ≤ r ≤ h′) is a generic splitting tower of q ⊗ E, then it is also a generic splitting tower of q. In particular h′ = h, i.e., h(q ⊗ E) = h(q). Moreover SP(q ⊗ E) = SP(q).

ii) If K/E is a generic splitting field of q ⊗ E of some level r ∈ [0, h], then K/k is a generic splitting field of q of the same level r.

Proof. i): This follows from the definition 1.12 of the notion of a generic splitting tower, together with 2.4. ii): We have K ∼_{E} E_{r}. This implies K ∼_{k} E_{r}, and we are done. □

2.9. Remark. Theorem 2.4 tells us that the splitting pattern of a quadratic form becomes coarser under base field extension. This may even happen with anisotropic k-forms, which stay anisotropic over the extension field L. The classical example is a quadratic form ψ of dimension 4 and with a non trivial discriminant (or Arf invariant) δψ. The form ψ remains anisotropic over the quadratic discriminant extension L = k_{δψ}. Of course the height of ψ_{L} is one, which means, that, over L, the form ψ is ‘simpler’ than over k. Such a transition ψ ↦ ψ_{L} is called an anisotropic splitting, since it reduces the complexity of the quadratic form ψ without disturbing its anisotropy. This phenomenon can be more subtle than in the example just given. For the rest of this remark we assume that char k ≠ 2.

i) For an an anisotropic r-Pfister form φ with pure part φ′, and ψ as above, we study the form q := φ′ ⊗ ψ. As mentioned above, ψ_{L} is anisotropic for L = k(\sqrt{δq}). We also assume that φ and hence φ′ as well as q stay anisotropic over L: For example, we can start with some ground field k_{0}, and let k = k_{0}(X_{1}, X_{2}, Y_{1}, … Y_{r}) be the function field of r + 2 indeterminates over k_{0}, and then take

ψ = ⟨1, X_{1}, X_{2}, δX_{1}X_{2}⟩ with δ ∈ k_{0}^{∗} \ k_{0}^{∗2}, φ = ⟨⟨Y_{1}, …, Y_{r}⟩⟩.

Then (φ ⊗ ψ)_{L} is an anisotropic r + 2-Pfister form, and q_{L} is a Pfister neighbor of that form with complement ψ_{L}. Hence q_{L} is an excellent form of height 2 with splitting pattern

SP(q_{L}) = (2^{r+1} − 4, 2^{r+1} − 2).

On the other hand, if E = k(\sqrt{−X_{1}}), then ψ_{E} = H ⊥ ψ_{E,an}, and hence q_{E} = (φ′ ⊗ ψ)_{E} = φ′_{E} ⊗ H ⊥ φ′_{E} ⊗ ψ_{E,an}. Using, e.g., [HR_{2}, 1.2, p. 165] one sees easily that φ′_{E} ⊗ ψ_{E,an} is anisotropic. It is similar to a Pfister neighbor with complement ψ_{E,an}, hence of height two with splitting pattern (2^{r} − 1, 2^{r+1} − 3). The splitting pattern of q therefore contains the numbers

2^{r} − 1, 2^{r+1} − 4, 2^{r+1} − 3, 2^{r+1} − 2,

and possibly more, but only two of them survive for q_{L}. ii) The following example may be even more instructive. We refer to [HR_{2}, 2.5– 2.10, p. 167ff.]. Assume n ≥ r > 0, and let k = k_{0}(X_{1}, …, X_{n}, Y_{1}, …, Y_{r}, Z be the function field over some field k_{0} in n + r + 1 indeterminates. We let k′ denote the subfield k_{0}(X_{1}, …, X_{n}, Y_{1}, …, Y_{r}) of k, hence k = k′(Z) is an inessential extension of k′. We consider the anisotropic forms

q := ⟨⟨X_{1}, …, X_{n}⟩⟩ ⊥ Z⟨⟨Y_{1}, …, Y_{r}⟩⟩  over k

and

ψ := ⟨⟨X_{1}, …, X_{n}⟩⟩ ⊥ ⟨⟨Y_{1}, …, Y_{r}⟩⟩  over k′.

According to [l.c., 2.6], their splitting pattern is given by

(∗)  \begin{matrix}(0, 2^{0}, 2^{1}, …, 2^{r}, 2^{n−1}, 2^{n−1} + 2^{r−1}) \\ (0, 2^{0}, 2^{1}, …, 2^{n−1}, 2^{n−1} + 2^{n−2}) \\ (0, 2^{0}, 2^{1}, …, 2^{n}) \end{matrix}  \begin{matrix}if 1 ≤ r ≤ n − 2 \\ if r = n − 1 \\ if r = n \end{matrix}

(The proof is given in [l.c.] for q, but works, mutatis mutandis, for ψ as well.) We consider the standard generic splitting tower K_{0} = k, K_{1}, …, K_{h} of ψ_{k}, which is a generic splitting tower of ψ as well, since k is inessential over k′, and note that h = r + 3, r + 2, r + 1 respectively in the three cases distinguished above.

The forms σ := ⟨⟨X_{1}, …, X_{n}⟩⟩_{K_{i}} and τ := ⟨⟨Y_{1}, …, Y_{r}⟩⟩_{K_{i}} are anisotropic for r = 0, …, r. Hence, using [l.c., Thm. 1.2, p. 165], we conclude that q_{K_{i}} is anisotropic for i = 0, …, r. In [l.c., 2.5, p. 167] the following well known linkage result is stated: If a linear combination of two anisotropic Pfister forms σ, τ over a given field is isotropic, then its index is the dimension of a Pfister form of maximal dimension dividing both σ and τ. Since ψ_{K_{i}} is isotropic, it follows that its index is a power of two, since it is the dimension of the common maximal Pfister divisor of σ and τ. Hence, by the same result, the first higher index of q_{K_{i}} is exactly the dimension of this Pfister divisor. Therefore, for i ≤ r, the splitting pattern of q_{K_{i}} consists of 0, followed by the suffix starting with 2^{i} of the appropriate sequence (∗). This shows that the gaps occurring in a splitting pattern by base field extension can be arbitrarily large, even for a form which stays anisotropic over the extension.

§3. Defining higher kernel forms over purely transcendental field extensions

3.1. Definition. Let L/K be a field extension and φ a form over L. If we have φ ≅ φ′_{L} = φ′ ⊗ L with some form φ′ over K, then we say that φ is definable over K (by φ′). We say that φ is defined over K (by φ′) if φ′ is unique up to isometry over K.

It is known for a field k of characteristic ≠ 2, that all higher kernels of a form q over k are defined over k if and only the form q is excellent [K_{3}, 7.14,p. 6], which is a very strong condition on that form: q is excellent if either dim q ≤ 3, or if q is a Pfister neighbor with excellent complement. E.g., ∑_{i=1}^{n} X_{i}^{2} is excellent over any field of characteristic ≠ 2. In this section, as before, q is a (regular quadratic) form over a field k. We want to prove the surprising fact, that, for a suitable generic splitting tower of q, every higher kernel form of q is definable over some finitely generated purely transcendental extension of k. The following lemma is well known, but we will need its precise statement as given here later on.

3.2. Lemma. Assume that q is anisotropic. Let L be a separable quadratic field extension of k, such that q_{L} is isotropic. Then

q ≅ α ⊥ β

for some regular quadratic forms α, β over k, such that α_{L} is hyperbolic and β_{L} is anisotropic. Let L = k[X]/(aX^{2} + X + b) with a ≠ 0, b ≠ 0 (which can always be achieved). Then α is divisible by [a, b]. More precisely, if i = ind (q_{L}), then there are pairwise orthogonal vectors y_{1}, …, y_{i} over k such that

α = [a, b] ⊗ ⟨q(y_{1}), …, q(y_{i})⟩.

In case char k ≠ 2, we may assume that L = k(\sqrt{δ}). Then, α is divisible by ⟨1, −δ⟩.

Proof. Let e ≠ 0 be an isotropic vector for q_{L}. We denote the image of X in L by θ. Then, for e = x+yθ, where x, y have coordinates in k, we obtain ^{4)} 0 = aq_{L}(e) =

4) We briefly write (x, y): = B_{q}(x, y) with B_{q} the bilinear form associated to q aq(x) + aq(y)θ^{2} + a(x, y)θ = aq(x) − bq(y) + (a(x, y) − q(y))θ, hence aq(x) = bq(y) and a(x, y) = q(y). Therefore (x, y) ≠ 0 and q(ξy + ηx) = (aξ^{2} + ξη + bη^{2})(x, y) for arbitrary ξ, η ∈ k, which gives a binary orthogonal summand of q of the requested type. If its complement is anisotropic over L we are done. Otherwise, an induction on dim q gives the general result, and the special result for char k ≠ 2 is obtained as usual by the substitution X = X′ − 1/(2a), δ = b/a − 1/(4a^{2}). □

3.3. Corollary. Let k(q) denote the function field of the quadric given by q = 0, let k′ ⊂ k(q) be a subfield containing k such that k(q)/k′ is separable quadratic and k′/k is purely transcendental. Then there is a decomposition

q_{k′} = α ⊥ q′,

over k′, such that α ≠ 0, α_{k(q)} is hyperbolic and q′_{k(q)} is anisotropic. Hence the first higher kernel form of q is definable by q′ over the purely transcendental extension k′ of k.

Proof. This follows immediately from 3.2. □

As before h = h(q) denotes the height of q.

3.4. Theorem. There exists a regular (cf. 2.3) generic splitting tower (K_{r} | 0 ≤ r ≤ h) of q, a tower of fields (K′_{r} | 0 ≤ r ≤ h) with k ⊂ K′_{r} ⊂ K_{r} for every r, k = K′_{0} = K_{0}, and a sequence (φ_{r} | 0 ≤ r ≤ h) of forms φ_{r} over K′_{r}, such that the following holds. {N.B. All the fields K_{r}, K′_{r} are subfields of K_{h}.}

(1) φ_{r} ⊗ K_{r} = ker(q ⊗ K_{r}) for every r ∈ [0, h].

(2) φ_{r+1} < φ_{r} ⊗ K′_{r+1} (0 ≤ r < h). ^{5)}

(3) K′_{r+1}/K′_{r} is purely transcendental of finite transcendence degree (0 ≤ r < h).

(4) K_{r}/K′_{r} is a finite multiquadratic extension (0 ≤ r ≤ h).

Proof. We proceed by induction on dim q. We may assume that q is anisotropic. If dim q ≤ 1 nothing has to be done. Assume now that dim q > 1. Let K_{1} = k(q), the function field of the projective quadric q = 0. We choose for K′_{1} a subfield of K_{1} containing k such that K′_{1}/k is purely transcendental and K_{1}/K′_{1} is quadratic, which is possible. By 3.3 we have a (not unique) decomposition q ⊗ K′_{1} = η_{1} ⊥ φ_{1} with dim φ_{1} < dim q, φ_{1} ⊗ K_{1} anisotropic, η_{1} ⊗ K_{1} hyperbolic. If the height h = 1, we have finished with K_{1}, K′_{1}, φ_{1}. Assume now that h > 1. We apply the induction hypothesis to φ_{1}. Let h(φ_{1}) = e. There exists a regular generic splitting tower (L_{j} | 0 ≤ j ≤ e) of φ_{1}, a tower of fields (L′_{j} | 0 ≤ j ≤ e), and forms ψ_{j} over L′_{j} (0 ≤ j ≤ e), such that K′_{1} ⊂ L′_{j} ⊂ L_{j}, ψ_{j+1} < ψ_{j} ⊗ L′_{j+1}, ψ_{j} ⊗ L_{j} = ker(φ_{1} ⊗ L_{j}), L′_{j+1}/L′_{j} is purely transcendental of finite degree, and L_{j}/L′_{j} is finite multiquadratic. Certainly e ≥ h − 1 ≥ 1 since h(φ_{1} ⊗ K_{1}) = h − 1. We form the field composites K_{1} · L_{j} = K_{1} ·_{K′_{1}} L_{j} as explained in 2.2. Let J denote the set of indices j ∈ [0, e] with ψ_{j} ⊗ K_{1} · L_{j} anisotropic, and let μ(0) < μ(1) < ⋯ < μ(t) be a list of these indices. (N.B. μ(0) = 0, μ(t) = e.) By 2.4 and 2.9 the sequence of fields (K_{1} · L_{μ(r)} | 0 ≤ r ≤ t) is a regular generic splitting tower of φ_{1}⊗K_{1}, hence of φ⊗K_{1}, and φ⊗(K_{1}·L_{μ(r)}) has the kernel form ψ_{μ(r)}⊗(K_{1}·L_{μ(r)}). Clearly the tower (K_{1} · L_{μ(r)} | 0 ≤ r ≤ t) is regular, and t = h − 1.

5) cf. Notations 1.0 For 2 ≤ i ≤ h we put K′_{i}: = L′_{μ(i−1)}, K_{i} = K_{1} · L_{μ(i−1)}, φ_{i}: = ψ_{μ(i−1)}. Adding to these fields and forms the fields K′_{1}, K_{1}, K_{0} = K′_{0} = k, and the forms φ_{1}, φ_{0}: = q, we have towers (K_{r} | 0 ≤ r ≤ h), (K′_{r} | 0 ≤ r ≤ h) and a sequence (φ_{r} | 0 ≤ r ≤ h) of anisotropic forms with all the properties listed in the theorem. □

We add to this theorem a further observation.

3.5. Proposition. We stay in the situation of Theorem 3.4. Assume that h ≥ 1, i.e., q is not split. By property (2) we have a sequence (η_{r} | 1 ≤ r ≤ h) of forms η_{r} over K′_{r} such that

φ_{r−1} ⊗ K′_{r} ≅ η_{r} ⊥ φ_{r} (1 ≤ r ≤ h).

We choose for each r ∈ [1, h] a total generic splitting field E_{r} of η_{r}. Let q_{r} denote the kernel form of q ⊗ K_{r}, 0 ≤ r ≤ h. Claim. The field composite K_{r−1} ·_{K′_{r−1}} E_{r} =: L_{r} is specialization equivalent to K_{r} over K_{r−1}. Thus L_{r} is a generic zero field of q_{r−1} and a generic splitting field of q of level r.

Proof. q_{r−1} ⊗ L_{r} is isotropic. Thus there exists a place λ: K_{r} → L_{r} ∪ ∞ over K_{r−1}. On the other hand η_{r} ⊗ K_{r} ∼ 0. Thus there exists a place ρ: E_{r} → K_{r} ∪ ∞ over K′_{r}. The field extension K_{r−1}/K′_{r−1} is finite multiquadratic. By standard valuation theoretic arguments ρ extends to a place μ: K_{r−1} ·_{K′_{r−1}} E_{r} → K_{r} ∪ ∞ over K_{r−1} · K′_{r}, hence over K_{r−1}. □

Explanations on the diagram. The tower on the left consists of purely transcendental extensions (labeled by “p.t.”), the tower on the right is a generic splitting tower of q. The “horizontal” extensions labeled by “m.q.” are multiquadratic,

splitting the direct sums ε_{r} = \underline{|}_{i=1}^{r−1} η_{i,K′_{i}} ⊥ η_{r} (for which we simply have written

η_{1} ⊥ ⋯ ⊥ η_{r}) totally and leaving φ_{r} anisotropic, making it isometric to the r-th higher kernel form q_{r} of q. The field E_{r} = K′_{r}(η_{r} → 0) is a generic total splitting field of the form η_{r} over K′_{r}.

A Generic Splitting Tower According To 3.4 and 3.5

[figure: K′_{h}; m.q.; η_{1}⊥···⊥η_{h} →0; φ_{h}→(q_{K_{h}})_{an}; K_{h}; E_{r} = K′_{r}(η_{r} → 0); E_{r} ∗_{K′_{r−1}}K_{r−1}; K′_{r}; m.q.; η_{1}⊥···⊥η_{r} → 0; φ_{r}→ (q_{Kr})_{an}; p.t.; η_{r}→0; K_{r}; K_{r−1}-equivalence; K′_{r−1}; m.q.; η_{1}⊥···⊥η_{r−1} → 0; φ_{r−1}→ (q_{K_{r−1}} )_{an}; p.t.; K_{r−1}; K′_{1}; m.q.; φ_{1} → (q_{K_{1}} )_{an}; η_{1} → 0; p.t.; K_{1} = K(q); k; p.t.; K_{0}]

§4. Generic splitting preparation

In 3.4 and 3.5 we have associated to the form q over k – among other things – a tower (K′_{r} | 0 ≤ r ≤ h) of purely transcendental field extensions of k together with a sequence (η_{r} | 1 ≤ r ≤ h) of anisotropic subforms η_{r} of q ⊗ K′_{r}. We want to understand in which way these data control the splitting behavior of q under field extensions, forgetting the generic splitting tower (K_{r} | 0 ≤ r ≤ h) in 3.4. (We will have only partial success, cf. §6 below.) In addition we strive for an abstraction of the situation established in 3.4 and 3.5. As before q is any regular quadratic form over a field k and h denotes the height h(q).

4.1. Definition. Let r ∈ [0, h] and let K/k be an inessential field extension (cf. 2.7). A form η over K is called a generic splitting form of q of level r, if dim η is even and there exists an orthogonal decomposition q ⊗ K ≅ η ⊥ ψ such that the following holds: If E/K is a total generic splitting field of η, then E/k is a partial generic splitting field of q of level r, while ψ ⊗ E is anisotropic. We further call ψ the complement (or complementary form) of η. □

N.B. This property does not depend on the choice of the total generic splitting field of E. Generic splitting forms occur whenever a higher kernel form of q is definable over an inessential field extension of k.

4.2. Proposition. Let K/k be a partial generic splitting tower of q of level r. Assume there is given an inessential subextension K′/k of K/k and a subform ψ of q ⊗ K′ such that ψ ⊗ K = ker(q ⊗ K). Let η denote the complement of ψ in q ⊗ K′, i.e., q ⊗ K′ ≅ η ⊥ ψ. (N.B. η is uniquely determined up to isometry.) Then η is a generic splitting form of q of level r.

Proof. dim q ≡ dim ψ mod 2. Thus dim η is even. Let E/K′ be a total generic splitting field of η. Then q ⊗ E ∼ ψ ⊗ E. Thus there exists a place λ: K → E ∪ ∞ over k. On the other hand, η ⊗K ∼ 0. Thus there exists a place μ: E → K ∪∞ over K′. Since μ is also a place over k, the fields E and K are specialization equivalent over k. We conclude that also E is a generic splitting field of q of level r. Now it is evident that dim ker(q ⊗ E) = dim ψ. Thus ψ ⊗ E is anisotropic. □

4.3. Definition. A generic splitting preparation of q is a tower of fields (K′_{r} | 0 ≤ r ≤ h) together with a sequence (η_{r} | 0 ≤ r ≤ h) of forms η_{r} over K′_{r} such that the following holds:

(1) K′_{0} = k, and η_{0} is the hyperbolic part of φ.

(2) K′_{r+1}/K′_{r} is purely transcendental for every r, 0 ≤ r < h. ^{6)}

(3) There exist orthogonal decompositions

q ≅ η_{0} ⊥ φ_{0}, φ_{r} ⊗ K′_{r+1} ≅ η_{r+1} ⊥ φ_{r+1}, (0 ≤ r < h).

(4) For every r ∈ [0, h] the form \underline{|}_{j=0}^{r} η_{j} ⊗ K′_{r} is a generic splitting form of q of

level r. □

The forms φ_{r} (0 ≤ r ≤ h) are uniquely determined (up to isometry, as always) by condition (3). We call φ_{r} the r-th residual form and η_{r} the r-th splitting form of the given generic splitting preparation. Clearly i_{r}: = dim η_{r}/2 is the r-th higher index (cf. 1.1) of q, and dim φ_{h} ≤ 1. If q is anisotropic then we sometimes denote the generic splitting preparation by (K′_{r} | 0 ≤ r ≤ h), (η_{r} | 1 ≤ r ≤ h), omitting the trivial form η_{0} = 0. □

4.4. Scholium. Generic splitting preparations of q exist in abundance. Indeed, in the situation described in 3.4 and 3.5, the tower of fields (K′_{r} | 0 ≤ r ≤ h) together with the sequence (η_{r} | 0 ≤ r ≤ h) of forms η_{r} over K′_{r}, where η_{r} for r ≥ 1 has been introduced in 3.5 and η_{0} denotes the hyperbolic part of q, is a generic splitting preparation of q.

6) Things below would not change much if we merely demanded that the extensions K′_{r+1}/K′_{r} are inessential.

Proof. Let r ∈ [1, h] be fixed and ε_{r}: = \underline{|}_{j=0}^{r} η_{j} ⊗ K′_{r}. Then q ⊗ K′_{r} ≅ ε_{r} ⊥ φ_{r} and

ker(φ ⊗ K_{r}) = φ_{r} ⊗ K_{r}. Proposition 4.2 tells us that ε_{r} is a generic splitting form of q of level r. □

Notice that in this argument property (4) of Theorem 3.4 has not been used. In the following, we study a fixed generic splitting preparation (K′_{r} | 0 ≤ r ≤ h), (0 ≤ η_{r} ≤ h) of q with associated residual forms φ_{r} (0 ≤ r ≤ h). For every

r ∈ \{1, …, h\} let ε_{r} denote the form \underline{|}_{j=0}^{r} η_{j} ⊗ K′_{r}. We do not assume that the

preparation arises in the way described in the scholium. Our next goal is to derive a generic splitting preparation of q ⊗ L from these data for a given field extension L/k.

4.5. Lemma. Assume that K/k is an inessential field extension. Let η be a form over K which is a generic splitting form of q, and let ψ be the complementary form, φ ⊗ K ≅ η ⊥ ψ. Let E/K be a regular total generic splitting field of η (cf. 2.2). Finally, let K · L = K ·_{k} L denote the free field composite of K and L over k, and E · L = E ·_{k} L the composite of E and L as explained in 2.2. Assume that ψ ⊗ E · L remains anisotropic. Then K · L/L is again inessential and η ⊗ K · L is a generic splitting form of q ⊗ L with complementary form ψ ⊗ K · L.

Proof. Any place α: K → L ∪ ∞ over k extends to a place from L · K to L over L. Thus K·L/L is inessential. By Proposition 2.6 the field E ·_{k} L = E ·_{K} (K ·_{k} L) is a total generic splitting field of η ⊗ K·L and also a partial generic splitting field of φ ⊗ L. Now the claim is obvious from Definition 4.1. □

As in §1 and §2 we enumerate the splitting pattern SP(q) = {j_{r} | 0 ≤ r ≤ h} by

0 ≤ j_{0} < j_{1} < ⋯ < j_{h} = [\frac{dim q}{2} ]

and write SP(q ⊗ L) = {j_{r} | r ∈ J} with J = {t(s) | 0 ≤ s ≤ e},

0 ≤ t(0) < t(1) < ⋯ < t(e) = h.

We have e = h(q ⊗ L).

4.6. Proposition. For 0 < s ≤ e we define L′_{s} = L · K′_{t(s)} as the free composite of the fields L and K′_{t(s)} over k, and we put

ζ_{s}: = η_{t(s−1)+1} ⊗ L′_{s} ⊥ ⋯ ⊥ η_{t(s)−1} ⊗ L′_{s} ⊥ η_{t(s)}.

We further define L′_{0} = L and ζ_{0} as the hyperbolic part of φ ⊗ L. Then (L′_{s} | 0 ≤ s ≤ e), (ζ_{s} | 0 ≤ s ≤ e) is a generic splitting preparation of q ⊗ L.

Proof. For every r with 0 < r ≤ h we choose a regular total generic splitting field F_{r}/K′_{r} of the form ε_{r}.^{∗)} ^{7)} Then F_{r}/k is a partial generic splitting field of q of level r. Let L · F_{r} = L ·_{k} F_{r} denote the composite of L with F_{r} over k as explained in

7) The fields F_{i} from 1.3 will not be used in the following.

2.2. If 0 < s ≤ e, then the field L · F_{t(s)} is a generic splitting field of φ ⊗ L and ker(φ ⊗ L · F_{t(s)}) = φ_{t(s)} ⊗ L · F_{t(s)} by Proposition 2.6. We have

φ ⊗ L′_{s} ≅ ε_{t(s)} ⊗ L′_{s} ⊥ φ_{t(s)} ⊗ L′_{s}.

Again by 2.6, the field

L ·_{k} F_{t(s)} = (L ·_{k} K′_{t(s)}) ·_{K′_{t(s)}} F_{t(s)} = L′_{s} ·_{K′_{t(s)}} F_{t(s)}

is a total generic splitting field of ε_{t(s)} ⊗L′_{s}. The form φ_{t(s)} ⊗L′_{s} remains anisotropic over L ·_{k} F_{t(s)}. Now Proposition 4.2 tells us that ε_{t(s)} ⊗ L′_{s} is a generic splitting form of φ ⊗ L′_{s}, and we are done. □

4.7. Corollary. Let m ∈ [1, h] be fixed, and let F be a regular total generic splitting field of ε_{m}. For every r with m < r ≤ h let F ·K′_{r} denote the free composite of F and K′_{r} over k. Then the tower F ⊂ F · K′_{m+1} ⊂ ⋯ ⊂ F · K′_{h} together with the sequence (η_{r} ⊗ F · K′_{r} | m < r ≤ h) is a generic splitting preparation of the anisotropic form φ_{m} ⊗ F.

Proof. We apply Proposition 4.6 with L = F. Now J = {m, m + 1, … h}. Thus e = h − m and t(s) = s + m (0 ≤ s ≤ h − m). The form q ⊗ F has the kernel form φ_{m} ⊗ F. □

We now look for generic splitting preparations with K′_{1} = ⋯ = K′_{h}. These are the “generic decompositions” according to the following definition.

4.8. Definition. Let K′ ⊃ k be a purely transcendental field extension. A generic splitting decomposition of q over K′ is a sequence (α_{0}, α_{1}, …, α_{h}) of forms over K′ of even dimensions such that the following holds.

i) q ⊗ K′ ≅ α_{0} ⊥ α_{1} ⊥ ⋯ ⊥ α_{h} ⊥ φ_{h} with dim φ_{h} ≤ 1.

ii) α_{0} is the hyperbolic part of q ⊗ K′. For every r with 1 ≤ r ≤ h the form α_{0} ⊥ ⋯ ⊥ α_{r} is a generic splitting form of q of level r. □

The next proposition tells us that we can always pass from a generic splitting preparation of q to a generic splitting decomposition of q.

4.9. Proposition. As before, let (K′_{r} | 0 ≤ r ≤ h), (η_{r} | 0 ≤ r ≤ h) be a generic splitting preparation of q. Then (η_{r} ⊗ K′_{h} | 0 ≤ r ≤ h) is a generic splitting decomposition of q over K′_{h}.

Proof. Let K′: = K′_{h} and α_{r}: = η_{r} ⊗ K′_{h} (0 ≤ r ≤ h). Since η_{0} is the hyperbolic part of q and K′/k is purely transcendental, the form α′_{0} is the hyperbolic part of q ⊗ K′. For 1 ≤ r ≤ h we have – with the notations from above – ε_{r} ⊗ K′ = α_{0} ⊥ ⋯ ⊥ α_{r}. Let F_{r} be a regular total generic splitting field of ε_{r} (over K′_{r}), hence a partial generic splitting field of q (over k) of level r. The free composite F_{r} · K′ = F_{r} ·_{K′_{r}} K′ is a total generic splitting field of ε_{r} ⊗ K′ = α_{r} by Proposition 2.6. Now F_{r} · K′ is purely transcendental over F_{r}. Thus F_{r} · K′ is also a partial generic splitting field of q of level r. We have q ⊗ K′ ≅ α_{r} ⊥ (φ_{r} ⊗ K′). The form φ_{r} ⊗ F_{r} is anisotropic. Since F_{r} · K′/F_{r} is purely transcendental, it follows that (φ_{r} ⊗K′)⊗F_{r} ·K′ = (φ_{r} ⊗F_{r})⊗F_{r} ·K′ is anisotropic. Thus α_{r} is a generic splitting form of q of level r. □

Although generic splitting decompositions look simpler than generic splitting preparations, it is up to now not clear to us which of the two concepts is better to work with. See also our discussion below at the end of §6.

§5. A brief look at quadratic places

We need some more terminology. If K is a field, we denote the group of square classes K^{∗}/K^{∗2} by Q(K) and a single square class aK^{∗2} by ⟨a⟩, identifying this class with the bilinear form ⟨a⟩ over K. If λ: K → L ∪ ∞ is a place with associated valuation ring 𝔬 = 𝔬_{λ}, we denote the image of the unit group 𝔬^{∗} in Q(K) by Q(𝔬). Notice that Q(𝔬) ≅ 𝔬^{∗}/𝔬^{∗2}, and that λ gives us a homomorphism λ_{∗}: Q(𝔬) → Q(L), λ_{∗}(⟨ε⟩) = ⟨λ(ε)⟩. (The bilinear form ⟨ε⟩ has good reduction under λ and λ_{∗}(⟨ε⟩) is the specialization of this form under λ.)

5.1. Definition. A quadratic place, or Q-place for short, from a field K to a field L is a triple (λ, H, χ) consisting of a place λ: K → L ∪ ∞, a subgroup H of Q(K) containing Q(𝔬_{λ}), and a homomorphism χ: H → Q(L) (called a “character” in the following) extending the homomorphism λ_{∗}: Q(𝔬_{λ}) → Q(L). □

We often denote such a triple (λ, H, χ) by a capital Greek letter Λ and symbolically write Λ: K → L ∪ ∞ for the Q-place Λ. Every place λ: K → L ∪ ∞ gives us a Q-place λ̂ = (λ, Q(𝔬_{λ}), λ_{∗}): K → L ∪ ∞, where λ_{∗}: Q(𝔬_{λ}) → Q(L) is defined as above. We regard λ and λ̂ essentially as the same object. In this sense Q-places are a generalization of the usual places.

5.2. Definition. If Λ = (λ, H, χ): K → L ∪ ∞ is a Q-place then an expansion of Λ is a Q-place Λ′ = (λ, H′, χ′): K → L ∪ ∞ with the same first component λ as Λ, a subgroup H′ of Q(K) containing H, and a character χ′: H′ → Q(L) extending χ. □

Usually Λ allows many expansions, and Λ itself is an expansion of λ̂. In the following Λ = (λ, H, χ): K → L ∪ ∞ a Q-place and 𝔬: = 𝔬_{λ}.

5.3. Definitions. Let k be a subfield of K.

a) The restriction Λ|k of the Q-place Λ to k is the Q-place (ρ, E, σ) where ρ = λ|k: k → L∪∞ denotes the restriction of the place λ: K → L∪∞ in the usual sense, E denotes the preimage of H in Q(k) under the natural map j: Q(k) → Q(K), and σ denotes the character χ ∘ (j|D) from D to Q(L).

b) If Γ: k → L ∪ ∞ is any Q-place from k to L then we say that Λ extends Γ (or, that Λ is an extension of Γ), if Λ|k is an expansion of Γ. □

At the first glance one might think that this notion of extension is not strong enough. One should demand Λ|k = Γ, in which case we say that Λ is a strict extension of Γ. But it will become clear below (cf. 5.8.ii, 5.12, 6.1) that the weaker notion of extension as above is the one needed most often. As before, we stay with a Q-place Λ = (λ, H, χ): K → L ∪ ∞.

5.4. Definition. Let φ be a regular quadratic form over K. We say that φ has good reduction (abbreviated: GR) under Λ, if there exists an orthogonal decomposition

(∗)  φ = \underline{|}_{h∈H} hψ_{h}

with forms ψ_{h} over K which all have GR under λ. Here hψ_{h} denotes the product ⟨h⟩ ⊗ ψ_{h} of the bilinear form ⟨h⟩ with ψ_{h}. {This amounts to scaling ψ_{h} by a representative of the square class ⟨h⟩.} Of course, ψ_{h} ≠ 0 for only finitely many h ∈ H.

Alternatively we say in this situation that φ is Λ-unimodular. In harmony with this speaking we call a form ψ over K, which has GR under λ, also a λ-unimodular form. □

5.5. Remark. We may choose a subgroup U of H such that H = U × Q(𝔬). If φ has GR under Λ then we can simplify the decomposition (∗) to a decomposition

φ ≅ \underline{|}_{u∈U} uφ_{u}

with λ-unimodular forms φ_{u}. □

5.6. Proposition. If φ has GR under Λ, and a decomposition (∗) is given, then the form \underline{|}_{h∈H} χ(h)λ_{∗}(ψ_{h}) is up to isometry independent of the choice of the decom-

position (∗).

This has been proved in [K_{4}] if charL ≠ 2. A proof in general (which is rather different) will be contained in [K_{5}].

5.7. Definition. If φ has GR under Λ then we denote the form \underline{|}_{h∈H} χ(h)λ_{∗}(ψ_{h})

(cf. 5.6) by Λ_{∗}(φ), and we call Λ_{∗}(φ) the specialization of φ under λ. □

5.8. Remarks.

i) If ψ is a second form over K such that φ ⊥ ψ is regular, and if both φ and ψ have GR under Λ then φ ⊥ ψ has GR under Λ and

Λ_{∗}(φ ⊥ ψ) ≅ Λ_{∗}(φ) ⊥ Λ_{∗}(ψ).

ii) Assume that k is a subfield of K and Γ: k → L ∪ ∞ is a Q-place such that Λ extends Γ (cf.5.3.b). Let q be a regular form over k which has GR under Γ. Then q ⊗ K has GR under Λ and Λ_{∗}(q ⊗ K) ≅ Γ_{∗}(q).

We omit the easy proofs. □ It seems that quadratic places come up in connection with generic splitting forms (cf. 4.1) in a natural way. We illustrate this by a little proposition, which will also serve us to indicate some of the difficulties we have to face if we want to make good use of quadratic places in generic splitting business.

5.9. Proposition. Let q be an anisotropic regular form over a field k, dim q > 2. Let k(q) denote the function field of the projective quadric q = 0. Let L ⊃ k be a field extension such that q_{L} = q ⊗ L is isotropic. Then there is a purely transcendental subextension k′/k of k(q)/k, such that k(q)/k′ is separable quadratic, and a quadratic place Λ: k′ → L ∪ ∞ over k, such that the form α described in 3.3 {q_{k′} ≅ α ⊥ q′, α_{k(q)} ∼ 0, q′_{k(q)} anisotropic} has GR under Λ and Λ_{∗}(α) ∼ 0.

Proof. Since q_{L} is isotropic we have a place λ: k(q) → L ∪ ∞ over k. Let 𝔬 denote the valuation ring of λ, let V denote the underlying k-vector space of q. We take a

decomposition V ≅ \underline{|}_{i=1}^{r} (ke_{i} ⊕ kf_{i}) ⊥ V′ with ke_{i} ⊕ kf_{i} = [a_{i}, b_{i}], and V′ = ⟨a_{0}⟩ or

V′ = 0, and a_{i}, b_{i}, a_{0} ∈ k^{∗}. We choose a primitive isotropic vector x ∈ V_{𝔬} = 𝔬 ⊗_{k} V with (x, V_{𝔬}) = 𝔬. By rearranging coordinates over k we may assume that x = Xe_{1} + Y f_{1} + z, and

X ∈ 𝔬 \ 0, Y ∈ 𝔬^{∗}, z ∈ \underline{|}_{i=2}^{r} (𝔬e_{i} ⊕ k𝔬f_{i}) ⊥ 𝔬V′, and after dividing by Y, we

may assume that Y = 1. We take the coordinates of \underline{|}_{i=2}^{r} (𝔬e_{i} ⊕ 𝔬f_{i}) ⊥ 𝔬V′ as

independent variables over k, which generate a purely transcendental subfield k′ of k(q). The equation 0 = q_{𝔬}(x) = a_{1}X^{2} + X + b with b = b_{1} + q_{k(q)}(z) ∈ 𝔬 ∩ k′ defines the field k(q) as a separable quadratic extension of k′. By construction we have λ(b) ≠ ∞. Hence the quadratic form [a_{1}, b] has good reduction under this place. We write α = [a_{1}, b] ⊗ ⟨c_{1}, …, c_{i}⟩ with c_{ν} ∈ ℑ(q_{k′}) \ 0 according to 3.2. We denote the restriction of λ to k′ by λ′, and the associated valuation ring by 𝔬′(= 𝔬 ∩ k′). Let H denote the subgroup of Q(k′) which is generated by Q(𝔬′) and the classes ⟨c_{1}⟩, …, ⟨c_{i}⟩. We choose some extension χ: H → Q(L) of the character λ′_{∗}: Q(𝔬′) → Q(L). The quadratic place Λ = (λ′, H, χ) has the desired properties. Alternatively we may choose for Λ the restriction of λ̂ to k′. □

This proposition leaves at least two things to be desired. Firstly, it would be nice and much more useful, to have the subextension k′/k to be chosen in advance, independently of the place λ in the proof. Secondly it would be pleasant if also the form q′ has GR under Λ. This is by no means guaranteed by our proof. We see no reason, why the analogue of Lemma 1.15.b for quadratic places instead of ordinary places should be true. The main crux here is the case char k = 2. Then usually many forms over k′ do not admit λ′-modular decompositions for a given place λ′: k′ → L ∪ ∞ over k. {It is easy to give counterexamples in a sufficiently general situation, cf. [K_{5}]}. On the other hand, if we can achieve in Proposition 5.9 in addition that q′ has GR under Λ, then the equation

q ⊗ L = Λ_{∗}(α) ⊥ Λ_{∗}(q′),

which follows from the remarks 5.8 above, together with Λ_{∗}(α) ∼ 0 would “explain” that q ⊗ L is isotropic, and how the anisotropic part of q ⊗ L is connected with the generic splitting form α of q of level 1. If char k ≠ 2 it is still not evident that the analogue of 1.15.b holds for the quadratic place Λ′ but now at least every form over k′ has a λ′-modular decomposition. Indeed, this trivially holds for forms of dimension 1, hence for all forms. In [K_{4}] a way has been found, to force an analogue of 1.15.b for quadratic places to be true, by relaxing the notion “good reduction” to a slightly weaker – but still useful – notion “almost good reduction”. We can define “almost good reduction” without restriction to characteristic ≠ 2 as follows.

5.10. Definition. Let Λ = (λ, H, χ): K → L ∪ ∞ be a Q-place, 𝔬: = 𝔬_{λ}, and let S be a subgroup of Q(K) such that Q(K) = S × H. A form φ over K has almost good reduction (abbreviated AGR) under Λ if φ has a decomposition

(∗∗)  φ ≅ \underline{|}_{s∈S} sφ_{s}

with Λ-unimodular forms φ_{s} and Λ_{∗}(φ_{s}) ∼ 0 for every s ∈ S, s ≠ 1. In this case we call the form

Λ_{∗}(φ): = Λ_{∗}(φ_{1}) ⊥ (dim φ − dim φ_{1})/2 × H

the specialization of φ under Λ. □

The point here is that Λ_{∗}(φ) is independent of the choice of the decomposition (∗∗) and also of the choice of S. This has been proved in [K_{4}] in the case char L ≠ 2. It also holds if char L = 2, cf. [K_{5}]. It now is almost trivial that the analogue of 1.15. holds for Q-places with AGR instead of GR, provided char L ≠ 2, cf. [K_{4}, §2].

5.11. Proposition. Let Λ: K → L ∪ ∞ be a Q-place and let φ and ψ be regular forms over K. Assume that char L ≠ 2.

a) If φ and ψ have AGR under Λ, then φ ⊥ ψ has AGR under Λ and

Λ_{∗}(φ ⊥ ψ) ≅ Λ_{∗}(φ) ⊥ Λ_{∗}(ψ).

b) If φ and φ ⊥ ψ have AGR under Λ, then ψ has AGR under Λ. □

5.12. Proposition. Let k ⊂ K be a field extension. Let Γ: k → L ∪ ∞ and Λ: K → L ∪ ∞ be Q-places with Λ extending Γ. Assume finally that q is a form over k with AGR under Γ. Then q ⊗ K has AGR under Λ and Λ_{∗}(q ⊗ K) ≅ Γ_{∗}(q).

The proof has been given in [K_{4}, §3] for char L ≠ 2. The arguments are merely book keeping. They remain true if char L = 2. □ Now we can repeat the arguments in the proof of Proposition 1.16 for quadratic places and AGR instead of usual places and GR, provided char L ≠ 2. We obtain the following.

5.13. Proposition. Let Λ: K → L ∪ ∞ be a Q-place and φ a form over K with AGR under Λ. Assume that char L ≠ 2. Then φ_{0}: = ker(φ) has again AGR under Λ and Λ_{∗}(φ) ∼ Λ_{∗}(φ_{0}), ind Λ_{∗}(φ) ≥ ind (φ). If ind Λ_{∗}(φ) = ind (φ), then ker Λ_{∗}(φ) = Λ_{∗}(φ_{0}). □

We finally state an important fact, proved in [K_{4}, §3], which has no counterpart on the level of ordinary places.

5.14. Proposition. Let again Λ: K → L∪∞ be a Q-place with char L ≠ 2. Let k be a subfield of K and Γ: = Λ|k. Let q be a regular form over k. Then q⊗L has AGR under Λ if and only if q has AGR under Γ, and in this case Λ_{∗}(q ⊗ L) ≅ Γ_{∗}(q). □

§6. Control of the splitting behavior by use of quadratic places

If we stay with fields of characteristic ≠ 2 then the propositions 5.11 – 5.13 indicate that it should be possible to obtain a complete analogue of the generic splitting theory displayed in §1 using quadratic places instead of ordinary places. Indeed such a theory has been developed in [K_{4}, §3]. We quote here the main result obtained there. Let q be a form over a field k. We return to some notations from §1: (K_{r} | 0 ≤ r ≤ h) is a generic splitting tower of q with higher indices (i_{r} | 0 ≤ r ≤ h) and higher kernel forms (φ_{r} | 0 ≤ r ≤ h). Further (j_{r} | 0 ≤ r ≤ h), with j_{r} = i_{0} + ⋯ + i_{r}, is the splitting pattern SP(q) of q.

6.1. Theorem. [K_{4}, Th. 3.7]. Let Γ: k → L ∪ ∞ be a Q-place into a field L of characteristic ≠ 2. Assume that q has AGR under Γ. We choose a Q-place Λ: K_{m} → L ∪ ∞ extending Γ such that either m = h or m < h and Λ does not extend to a Q-place from K_{m+1} to L. Then ind (Γ_{∗}(q)) = j_{m}, the form φ_{m} has GR under Λ and ker(Γ_{∗}(q)) ≅ Λ_{∗}(φ_{m}). □

A small point here – which we will not really need below – is that φ_{m} has GR under Λ, not just AGR.

6.2. Remark. The theorem shows that the generic splitting tower (K_{r} | 0 ≤ r ≤ h) “controls” the splitting behavior of Γ_{∗}(q). Indeed, if L′/L is any field extension then we obtain from Γ a Q-place j ∘ Γ: k → L′ ∪ ∞ in a rather obvious way (cf. 6.4.iii below). The form q has also AGR under j ∘ Γ, and (j ∘ Γ)_{∗}(q) = Γ_{∗}(q) ⊗ L′. We can apply the theorem to j∘Γ and q instead of Γ and q. In particular we see that ind (Γ_{∗}(q)⊗L′) is one of the numbers j_{r}. Thus the splitting pattern SP(Γ_{∗}(q)) is a subset of SP(q).□

We now aim at a result similar to Theorem 6.1, where the field K_{m} is replaced by an arbitrary partial generic splitting field for q. (This is not covered by [K_{4}].) For that reason we briefly discuss the “composition” of Q-places.

6.3. Definition. Let Λ = (λ, H, χ): K → L ∪ ∞ and M = (μ, D, ψ): L → F ∪ ∞ be Q-places. The composition M ∘ Λ of M and Λ is the Q-place

(M ∘ Λ) = (μ ∘ λ, H_{0}, ψ ∘ (χ|H_{0}))

with H_{0}: = {α ∈ H | χ(α) ∈ D}. □

6.4. Remarks.

i) If N: F → E ∪ ∞ is a third Q-place then N ∘ (M ∘ Λ) = (N ∘ M) ∘ Λ, as is easily checked.

ii) Let i: k ↪ K be a field extension, regarded as a trivial place. This gives us a “trivial” Q-place î = (i, Q(k), i_{∗}: Q(k) → Q(K)) from k to K. If Λ = (K, H, χ): K → L ∪ ∞ is any Q-places starting at K, then Λ ∘ î is the restriction Λ|k: k → L ∪ ∞.

iii) The Q-place j ∘ Γ alluded to in 6.2 is ĵ ∘ Γ. □

In all the following Γ: k → L ∪ ∞ is a Q-place into a field of characteristic ≠ 2 such that q has AGR under Γ.

6.5. Proposition. Let F/k be a generic splitting field of the form q of some level r ∈ [0, h]. Assume that ind Γ_{∗}(q) ≥ j_{r}. Then there exists a Q-place Λ: F → L ∪ ∞ extending Γ. For any such Q-place Λ the anisotropic part φ = ker(q⊗F) of q⊗F has AGR under Λ and Λ_{∗}(φ) ∼ ker Γ_{∗}(q). if ind Γ_{∗}(q) = j_{r}, then Λ_{∗}(φ) = ker Γ_{∗}(q).

Proof. We only need to prove the existence of a Q-place Λ: F → L ∪ ∞ extending Γ. The other statements are clear from §5 (cf. 5.12, 5.13). By Theorem 6.1 we have a Q-place Λ′: K_{r} → L ∪ ∞ extending Γ. We further have a place ρ: F → K_{r} over k. It now can be checked in a straightforward way that the Q-place Λ = Λ′ ∘ ρ̂ from F to L extends Γ. □

6.6. Theorem. Let K/k be an inessential field extension and q ⊗ K ≅ η ⊥ φ with η a generic splitting form of q of some level r ∈ [0, h]. Assume that ind Γ_{∗}(q) ≥ j_{r}. Then there exists a Q-place Λ: K → L ∪ ∞ extending Γ such that η has AGR under Λ and Λ_{∗}(η) ∼ 0. For every such Q-place Λ the form φ has AGR under Λ and Λ_{∗}(φ) ∼ Γ_{∗}(q). If ind Γ_{∗}(q) = j_{r}, then Λ_{∗}(φ) = ker Γ_{∗}(q).

Proof. Again it suffices to prove the existence of a Q-place Λ extending Γ such that η has AGR under Λ and Λ_{∗}(η) ∼ 0, the other statements being covered by §5 (cf. 5.11, 5.12). Let E be a total generic splitting field of η. Then E is a partial generic splitting field of q (cf. 4.1). By the preceding proposition 6.5 there exists a Q-place M: E → L∪∞ extending Γ. Let Λ: K → L ∪ ∞ denote the restriction of M to K (cf. 5.3). Of course, also Λ extends Γ. The form η ⊗ E is hyperbolic, hence certainly has AGR under M and M_{∗}(η ⊗ E) ∼ 0. Now Proposition 5.14 tells us that η has AGR under Λ and Λ_{∗}(η) ∼ 0. □

6.7. Remark. Suppose that Γ = γ̂ with γ: k → L ∪ ∞ a place and that q has GR under γ. Then Theorem 6.1 and Proposition 6.5 give us nothing more than we know from the generic splitting theory in §1. Indeed, if Λ = (λ, H, χ) is a Q-place as stated there, then the form φ_{m} in 6.1, resp. φ in 6.5, automatically has GR under λ. This is different with Theorem 6.6. Even in the case Γ = ĵ for j: k ↪ L the inclusion map into an overfield L of k (actually the case which is perhaps the most urgent at present), the Q-place Λ will be different from λ̂. Thus, in 6.6, Q-places instead of usual places are needed even in the case Γ = ĵ. □

We now choose again a generic splitting preparation (K′_{r} | 0 ≤ r ≤ h), (η_{r} | 0 ≤ r ≤

h) of q with residual forms φ_{r} and generic splitting forms ε_{r} = \underline{|}_{j=0}^{r} η_{j} ⊗ K′_{j} (0 ≤

r ≤ h), cf. 4.3.

6.8. Discussion. Assume that ind Γ_{∗}(q) ≥ j_{m} for some m ∈ [0, h]. Theorem 6.6 tells us that there exists a Q-place Λ: K′_{m} → L ∪ ∞ extending Γ such that ε_{m} has AGR under Λ and Λ_{∗}(ε_{m}) ∼ 0. Moreover, for every such Q-place λ the form φ_{m} has AGR under Λ and Λ_{∗}(φ_{m}) ∼ Γ_{∗}(q). If ind Γ_{∗}(q) = j_{m}, it follows that Λ_{∗}(φ_{m}) is the anisotropic part of Γ_{∗}(q). Assume now that ind Γ_{∗}(q) > j_{m} and Λ: K′_{m} → L∪∞ is a Q-place as just described. Then by analogy with 1.18 one might suspect at first glance that Λ extends to a place M: K′_{m+1} → L ∪ ∞ such that ε_{m+1} has AGR under M and M_{∗}(ε_{m+1}) ∼ 0. {N.B. Since Λ_{∗}(ε_{m}) ∼ 0, this is equivalent to the property that η_{m+1} has AGR under M and M_{∗}(η_{m+1}) ∼ 0.} But this is too much to be hoped for. Indeed, let us consider the special case that K′_{1} = ⋯ = K′_{h} =: K′, i.e., we are given a generic splitting preparation (η_{0}, η_{1}, …, η_{h}) over K′. If Λ: K′ = K′_{m} → L ∪ ∞ is as above, then we can expand Λ = (λ, H, χ) to a Q-place Λ′ = (λ, Q(K′), ψ′): K′ → L ∪ ∞. Also Λ′ extends Γ, further ε_{m} has AGR under Λ′ and Λ′_{∗}(ε_{m}) = Λ_{∗}(ε_{m}) ∼ 0 (cf. 5.12). But since K′_{m+1} = K′, the only extension of Λ′ to K′_{m+1} is Λ′ itself, and there is no reason why η_{m+1} should have AGR under Λ′ and Λ′_{∗}(η_{m+1}) ∼ 0. □

Nevertheless, if ind Γ_{∗}(q) > j_{m+1}, there exists a somewhat natural procedure to obtain from a Q-place Λ: K′_{m} → L ∪ ∞ as above a Q-place M: K′_{m+1} → L ∪ ∞ extending Γ such that both ε_{m} ⊗ K′_{m+1} and ε_{m+1} have AGR under M and we have M_{∗}(ε_{m} ⊗ K′_{m+1}) ∼ 0 as well as M_{∗}(ε_{m+1}) ∼ 0. This runs as follows.

6.9. Procedure. Assume that ind Γ_{∗}(q) > j_{m}. Let Λ: K′_{m} → L∪∞ be a Q-place extending Γ such that ε_{m} has AGR under Λ and Λ_{∗}(ε_{m}) ∼ 0. We choose a regular total generic splitting field F of ε_{m}. By Proposition 6.5 the Q-place Λ extends to a Q-place Λ̃: F → L ∪ ∞. The form φ_{m} ⊗ F is anisotropic. We now invoke Corollary 4.7, which tells us that the tower F ⊂ F · K′_{m+1} ⊂ ⋯ ⊂ F · K′_{h} together with the sequence of forms (η_{r} ⊗ F · K′_{r} | m < r ≤ h) is a generic splitting preparation of φ_{m} ⊗ F. Here F · K′_{r} denotes the free composite of the fields F and K′_{r} over k. In particular η_{m+1} ⊗ F · K′_{r} is a generic splitting form of φ_{m} ⊗ F over F · K′_{r} of level 1. We have Λ̃_{∗}(φ_{m} ⊗ F) ∼ Λ̃_{∗}(q ⊗ F) = Γ_{∗}(q). Since ind Γ_{∗}(q) > j_{m}, we conclude that Λ̃_{∗}(φ_{m} ⊗ F) is isotropic. Thus, again by Proposition 6.5, Λ̃ extends to a Q-place M̃: F · K′_{m+1} → L ∪ ∞ such that η_{m+1} ⊗ F · K′_{m+1} has AGR under M̃ and M̃_{∗}(η_{m+1} ⊗ F · K′_{m+1}) ∼ 0. Let M: K′_{m+1} → L ∪ ∞ denote the restriction of M̃ to K′_{m+1}. By 5.14, the form η_{m+1} has AGR under M and M_{∗}(η_{m+1}) ∼ 0. Now we need a delicate argument to prove that also ε_{m} ⊗ K′_{m+1} has AGR under M and M_{∗}(ε_{m} ⊗ K′_{m+1}) ∼ 0. The problem is that the diagram of field embeddings

\xymatrix{F \ar[r] & F · K′_{m+1} \\ K′_{m} \ar[u] \ar[r] & K′_{m+1} \ar[u]}

does not commute, since the field composite F · K′_{m+1} is built over k instead of K′_{m}. Thus M probably does not extend the Q-place Λ. The argument runs as follows. Also φ_{m+1} ⊗ F · K′_{m+1} has AGR under M̃, hence φ_{m+1} has AGR under M, and

M_{∗}(φ_{m+1}) ≅ M̃_{∗}(φ_{m+1} ⊗ F · K′_{m+1}) ∼ Λ̃_{∗}(φ_{m} ⊗ F) ∼ Λ̃_{∗}(q ⊗ F) ≅ Γ_{∗}(q).

Since q ⊗ K′_{m+1} ≅ ε_{m} ⊗ K′_{m+1} ⊥ η_{m+1} ⊥ φ_{m+1} and both, η_{m+1} and φ_{m+1}, have AGR under M, also ε_{m} ⊗ K′_{m+1} has AGR under M, cf. 5.11, and

Γ_{∗}(q) ≅ M_{∗}(q ⊗ K′_{m+1}) ≅ M_{∗}(ε_{m} ⊗ K′_{m+1}) ⊥ M_{∗}(η_{m+1}) ⊥ M_{∗}(φ_{m+1}).

Since M_{∗}(η_{m+1}) ∼ 0 and M_{∗}(φ_{m+1}) ∼ Γ_{∗}(q), we conclude that M_{∗}(ε_{m} ⊗ K′_{m+1}) ∼ 0. □

We have ind Γ_{∗}(q) = j_{r} for some r ∈ [0, h] (cf. 6.2). Iterating the procedure with m = 0, 1, …, r − 1, we obtain the following theorem.

6.10. Theorem. Let ind Γ_{∗}(q) = j_{r}. Then there exists a Q-place Λ: K′_{r} → L ∪ ∞ extending Γ such that η_{m} ⊗ K′_{r} has AGR under Λ and Λ_{∗}(η_{m} ⊗ K′_{r}) ∼ 0 for every m ∈ [0, r]. If Λ is any such Q-place then φ_{r} has AGR under Λ and Λ_{∗}(φ_{r}) = ker Γ_{∗}(q).

We briefly discuss the case that L is a field extension of k and Γ = ĵ with j: k ↪ L the inclusion map.

6.11. Definition/Remark. Let K and L be field extensions of k. A Q-place from K to L over k is a Q-place Λ = (λ, H, χ): K → L∪∞ such that the first component λ is a place over k. It is evident from Definitions 5.3 that this condition just means that Λ extends the quadratic place ĵ: k → L ∪ ∞, and also that Λ|k = ĵ. □

6.12. Scholium. Let L/k be a field extension, ind (q ⊗ L) = j_{r}. Then there exists a Q-place Λ: K′_{r} → L ∪ ∞ over k such that η_{m} ⊗ K′_{r} has AGR under Λ and Λ_{∗}(η_{m} ⊗ K′_{r}) ∼ 0 for every m ∈ [0, r]. For any such Q-place Λ the form φ_{r} has AGR under Λ and Λ_{∗}(φ_{r}) = ker(q ⊗ L). □

We return to an arbitrary Q-place Γ: k → L ∪ ∞ such that q has AGR under Γ.

6.13. Definition. We call the generic splitting preparation (K′_{r} | 0 ≤ r ≤ h), (η_{r} | 0 ≤ r ≤ h) of q tame, if there exists a generic splitting tower (K_{r} | 0 ≤ r ≤ h) of q such that K′_{r} is a subfield of K_{r} and η_{r} ⊗ K_{r} ∼ 0 for every r ∈ [0, h]. □

Notice that a generic splitting preparation (K′_{r} | 0 ≤ r ≤ h), (η_{r} | 0 ≤ r ≤ h) as described by 3.4 and 3.5 is tame. Thus every form admits tame generic splitting preparations. On the other hand we suspect that there exist generic splitting preparations which are wild (= not tame), although we did not look for examples. If our given generic splitting preparation of q is tame then there exists a much simpler procedure than the one described above, to obtain a Q-place Λ: K′_{r} → L∪∞ with the properties stated in Theorem 6.10.

6.14. Procedure. Assume that ind Γ_{∗}(q) = j_{r}, and that (K_{i} | 0 ≤ i ≤ h) is a generic splitting tower of q such that K′_{i} is a subfield of K_{i} and ε_{i} ⊗ K_{i} ∼ 0 for every i ∈ [0, h]. By 6.1 there exists a Q-place Λ̃: K_{r} → L ∪ ∞ extending Γ. Let Λ denote the restriction of Λ̃ to K′_{r}. Then Λ again extends Γ. For every i ∈ [0, r] we have (ε_{i} ⊗ K′_{r}) ⊗ K_{r} = ε_{i} ⊗ K_{r} ∼ 0. Certainly ε_{i} ⊗ K_{r} has AGR under Λ̃ and Λ̃_{∗}(ε_{i} ⊗ K_{r}) ∼ 0. By 5.14 the form ε_{i} ⊗ K′_{r} has AGR under Λ and Λ_{∗}(ε_{i} ⊗ K′_{r}) ∼ 0. Since this holds for every i ∈ [0, r] we conclude (using 5.11) that η_{i} ⊗ K′_{r} has AGR under Λ and Λ_{∗}(η_{i} ⊗ K′_{r}) ∼ 0 for every i ∈ [0, r].

Thus it seems that life is easier if we have a tame generic splitting preparation at our disposal than an arbitrary one. Up to now this is an argument in favor of working with generic splitting preparations instead of the more special generic splitting decompositions (cf. 4.8), in spite of Proposition 4.9, for we do not know whether every form admits a tame generic splitting decomposition.

References

[A] C. Arf, Untersuchungen über quadratische Formen in Körpern der Charakteristik 2 (Teil I). J. reine angew. Math. 183 (1941), 148–167.

[HR_{1}] J. Hurrelbrink and U. Rehmann, Splitting patterns of excellent quadratic

forms, J. reine angew. Math. 444 (1993), 183–192. [HR_{2}] J. Hurrelbrink and U. Rehmann, Splitting patterns and linear combinations

of two Pfister forms. J. reine angew. Math. 495 (1998), 163–174 [J] N. Jacobson, “Lectures in abstract algebra” III, Theory of fields and Galois

theory, New York, Heidelberg, Berlin: Springer 1964.

[KR] I. Kersten and U. Rehmann, Generic splitting of reductive groups, Tôhoku Math. J. 46 (1994), 35–70.

[K_{1}] M. Knebusch, Specialization of quadratic forms and symmetric bilinear

forms, and a norm theorem, Acta Arithmetica, 24 (1973), 279–299. [K_{2}] M. Knebusch, Generic splitting of quadratic forms, I, Proc. London Math.

Soc. 33 (1976), 65–93. [K_{3}] M. Knebusch, Generic splitting of quadratic forms, II, Proc. London Math.

Soc. 34 (1977), 1–31. [K_{4}] M. Knebusch, Generic splitting of quadratic forms in the presence of bad

reduction, J. reine angew. Math. 517 (1999), 117–130. [K_{5}] M. Knebusch, “Spezialisierung von quadratischen und symmetrischen bilin-

earen Formen”, Book in preparation.

[Lg] S. Lang, “Introduction to algebraic geometry”. New York: Interscience 1958.

[W] E. Witt, Theorie der quadratischen Formen in beliebigen Koerpern. J. reine angew. Math. 176 (1937), 31–44.

E-mail address: manfred.knebusch@mathematik.uni-regensburg.de
               rehmann@mathematik.uni-bielefeld.de