Virtual forms

Karim Johannes Becher

Abstract

In this article, quadratic forms over a field of characteristic different from two are generalised to so-called virtual forms over an arbitrary field. These objects turn out to be useful for the study of the Milnor K-theory of a field. The relation between both areas is established by a sequence of maps corresponding to Delzant’s Stiefel-Whitney classes.

1 Introduction

Quadratic forms have manifold aspects and therefore they can be generalised in various directions. The aim of this article is to transfer some of the algebraic theory of quadratic forms over fields into a more general context. To this aim virtual forms over an arbitrary field are introduced. These objects are are closely related to Milnor K-theory.

To give the idea let us consider a field F of characteristic different from 2. The algebraic theory of quadratic forms over F, as it may be learned from [6] or [10], starts with a few crucial observations. First of all, quadratic forms over F can be diagonalised. Witt’s Cancellation Law then gives information on the interplay between orthogonal sum and isometry. This leads to the definition of the Witt ring W (F) and the Witt-Grothendieck ring Ŵ(F). Using further Witt’s Chain Equivalence Theorem, which describes when exactly two given diagonalisations belong to the same quadratic form, one obtains a description of W (F) and of Ŵ(F) by means of generators and relations.

This abstract description of the Witt-Grothendieck ring Ŵ(F) will be taken as the guiding principle for the definition of a group G(F, ℓ), the Grothendieck group of ℓ-forms, where now F is an arbitrary field and ℓ ∈ ℕ. If F is of characteristic different from 2, then G(F, 2) coincides with the group Ŵ(F). The definition of G(F, ℓ) is based on Milnor K-theory. After defining several operations on the elements of G(F, ℓ), a descending sequence of subgroups (G^{n}(F, ℓ))_{n∈ℕ} is constructed, destined to replace the filtration given by the powers of the fundamental ideal in the Witt-Grothendieck ring.

While in general G(F, ℓ) is not a commutative group for ℓ ≠ 2, at least the groups G^{n}(F, ℓ) for n ≥ 2 are contained in the center of G(F, ℓ). Moreover, if ℓ is odd, then G^{n}(F, ℓ) vanishes for n ≥ 3 and any element of G^{1}(F, ℓ) has finite order dividing ℓ. A sequence of maps from G(F, ℓ) to the Milnor Kgroups modulo ℓ of the field F allows to use ℓ-forms to investigate the Milnor K-theory of F. These maps generalise Delzant’s Stiefel-Whitney classes (in the way Milnor defines them in [8]). In certain cases, in particular for odd ℓ, it turns out that the group G^{1}(F, ℓ) can be described as a group extension of the first by the second Milnor K-group modulo ℓ. As an application of some of the new concepts an alternative proof of a result in K-theory due to B. Kahn is given. Finally, a notion of isotropy is introduced.

Part of the results presented here have been announced in [1].

Throughout this article F denotes a field, F^{×} its multiplicative group, and ℓ a nonnegative integer.

2 Milnor K-theory

Let us recall the basic definitions from Milnor K-theory (cf. [8]) and fix some notation. Let n, ℓ ∈ ℕ. Let K_{n}^{(ℓ)}F denote the nth K-group modulo ℓ of the Milnor K-theory of F. To be explicite, K_{n}^{(ℓ)}F is the additive abelian group which is generated by so-called symbols {a_{1}, …, a_{n}}, where a_{1}, …, a_{n} ∈ F^{×}, which are subject to the following relations:

(M1) the natural map { } : (F^{×})^{n} ⟶ K_{n}F is ℤ-multilinear;

(M2) {a_{1}, …, a_{n}} = 0 whenever a_{i} + a_{i+1} = 1 in F for some i < n;

(M3) ℓ · K_{n}^{(ℓ)}F = 0 .

Obviously (M3) can be ignored if ℓ = 0. In fact, K_{n}^{(0)}F is just the ‘full’ Milnor K-group usually denoted by K_{n}F, and K_{n}^{(ℓ)}F corresponds to its quotient modulo ℓ. Note that K_{1}^{(ℓ)}F is just the group F^{×}/F^{×ℓ} in additive notation if one identifies the element {x} of K_{1}^{(ℓ)}F with the class xF^{×ℓ} in F^{×}/F^{×ℓ}. For n = 0 the first two relations have no relevance, so K_{0}^{(ℓ)}F is a cyclic group generated by the empty symbol {}, and in view of (M3) we identify K_{0}^{(ℓ)}F = ℤ/ℓℤ. In K_{2}^{(ℓ)}F, it follows from (M1) and (M2) that {−a, a} = 0 and {a, b} = −{b, a} = {b^{−1}, a} hold for a, b ∈ F^{×} (cf. [8, §1]).

We denote K_{∗}^{(ℓ)}F = ⨁_{i≥0} K_{i}^{(ℓ)}F and K̂_{∗}^{(ℓ)}F = ∏_{i≥0} K_{i}^{(ℓ)}F . With

{a_{1}, …, a_{r}} · {a_{r+1}, …, a_{r+s}} = {a_{1}, …, a_{r}, a_{r+1}, …, a_{r+s}}

one has a natural multiplication in both K_{∗}^{(ℓ)}F and K̂_{∗}^{(ℓ)}F. Hence K_{∗}^{(ℓ)}F and K̂_{∗}^{(ℓ)}F are endowed with a natural structure as algebras over the ring K_{0}^{(ℓ)}F = ℤ/ ℓℤ, and K_{∗}^{(ℓ)}F is a graded algebra.

3 Chain equivalence

We consider finite sequences of elements in F^{×}. We denote by S_{n}(F) the set of all such sequences of length n and by [a_{1}, …, a_{n}] the sequence with entries a_{1}, …, a_{n} ∈ F^{×}. We use the sign ⋄ to denote the concatenation of two sequences: for α = [a_{1}, …, a_{r}] ∈ S_{r}(F) and α′ = [a_{r+1}, …, a_{r+s}] ∈ S_{s}(F) we set α ⋄ α′ = [a_{1}, …, a_{r}, a_{r+1}, …, a_{r+s}] ∈ S_{r+s}(F).

We introduce now, for any n ≥ 1, an equivalence relation on S_{n}(F) which depends on the (fixed) integer ℓ ≥ 0. This relation is introduced first for n = 1 and n = 2 and then extended in a canonical way to arbitrary n.

Let a, a′ ∈ F^{×}. We consider the sequences [a] and [a′] in S_{1}(F) to be equivalent in this relation (depending on ℓ) and write [a] ∼^{ℓ} [a′] in case that {a} = {a′} holds in K_{1}^{(ℓ)}F, that is, if a′ = x^{ℓ}a for some x ∈ F^{×}. Let now a, b, a′, b′ ∈ F^{×}. We write

[a, b] ∼^{ℓ} [a′, b′] if \{ \begin{matrix}\{ab\} = \{a′b′\} & in K_{1}^{(ℓ)}F and \\ \{a, b\} = \{a′, b′\} & in K_{2}^{(ℓ)}F . \end{matrix}

We thus have defined an equivalence relation on S_{n}(F) for n = 1, 2.

Let now n > 2. We say that the two sequences α = [a_{1}, …, a_{n}] and β = [b_{1}, …, b_{n}] in S_{n}(F) are simply ℓ-equivalent and write α ≈^{ℓ} β if there exists a positive integer k < n such that [a_{k}, a_{k+1}] ∼^{ℓ} [b_{k}, b_{k+1}] and such that we have {a_{i}} = {b_{i}} in K_{1}^{(ℓ)}F for 1 ≤ i < k and for k + 1 < i ≤ n. This relation is not transitive in general. In order to obtain an equivalence relation ∼^{ℓ} we take the transitive closure of ≈^{ℓ}. We say that α, β ∈ S_{n}(F) are chain ℓ-equivalent and write α ∼^{ℓ} β if

α ≈^{ℓ} γ_{1} ≈^{ℓ} γ_{2} ≈^{ℓ} … ≈^{ℓ} γ_{r} ≈^{ℓ} β

holds for a suitable choice of γ_{1}, …, γ_{r} ∈ S_{n}(F), r ≥ 1. For ℓ = 0 we just speak of simple equivalence and chain equivalence and we write ∼ and ≈, accordingly; note that simple (resp. chain) equivalence implies simple (resp. chain) ℓ-equivalence modulo any ℓ > 0.

The definition of chain ℓ-equivalence is motivated by a fundamental result in the algebraic theory of quadratic forms, Witt’s Chain Equivalence Theorem (cf. [6, Chap. I, §5]), which can be reformulated in the following way.

3.1 Proposition. Assume that char(F) ≠ 2. Two sequences [a_{1}, …, a_{n}] and [b_{1}, …, b_{n}] in S_{n}(F) are chain 2-equivalent if and only if the quadratic forms a_{1}X_{1}^{2} + … + a_{n}X_{n}^{2} and b_{1}X_{1}^{2} + … + b_{n}X_{n}^{2} over F are isometric.

Proof: For n = 1 the statement is obvious. Now we inspect the case n = 2.

For any c, d ∈ F^{×}, the symbols {cd} ∈ K_{1}^{(2)}F and {c, d} ∈ K_{2}^{(2)}F are invariants of the quadratic form cX^{2} + dY^{2} up to isometry. Hence, if the quadratic forms a_{1}X^{2} + a_{2}Y^{2} and b_{1}X^{2} + b_{2}Y^{2} are isometric, then we have

[a_{1}, a_{2}] ∼^{2} [b_{1}, b_{2}].

Conversely, assume that [a_{1}, a_{2}] ∼^{2} [b_{1}, b_{2}]. Since {a_{1}, a_{2}} = {b_{1}, b_{2}} holds in K_{2}^{(2)}F, the 3-dimensional quadratic forms −a_{1}X^{2} − a_{2}Y^{2} + a_{1}a_{2}Z^{2} and −b_{1}X^{2} − b_{2}Y^{2} + b_{1}b_{2}Z^{2} are isometric, by [3, Theorem 1.8] and Witt Cancellation (cf. [6, Chap. I, 4.2]). Since in addition we have b_{1}b_{2} = a_{1}a_{2}x^{2} for some x ∈ F^{×}, using Witt Cancellation and multiplying by −1 we see that a_{1}X^{2} + a_{2}Y^{2} and b_{1}X^{2} + b_{2}Y^{2} are isometric.

So far we have shown that 2-equivalence for elements in S_{2}(F) characterises isometry for the corresponding diagonal quadratic forms in two variables over F. Now, for arbitrary n ≥ 2, by Witt’s Chain Equivalence Theorem (cf. [6, Chap. I, §5]) isometry for quadratic diagonal forms in n variables is entirely determined by isometry for 2-dimensional diagonal subforms. This completes the proof. □

3.2 Remark. Assume that ℓ ≥ 2 and that F contains a primitve ℓth root of unity ζ. In this case, the Merkurjev-Suslin Theorem (cf. [7]) implies that symbols in K_{2}^{(ℓ)}F are in one-to-one correspondence with isomorphism classes of cyclic algebras of degree ℓ over F. More precisely, given a, b ∈ F^{×}, the symbol {a, b} is identified with the symbol algebra (a, b)_{F,ζ}. By definition, (a, b)_{F,ζ} is the central simple F -algebra of degree ℓ generated by elements u, v which are subject to the relations u^{ℓ} = a, v^{ℓ} = b, and vu = ζuv.

Therefore, under the assumptions on ℓ and F, chain ℓ-equivalence can be described in terms of classical algebraic structures not involving K-theory.

Permuting the entries of a sequence does not necessarily yield a sequence which is chain equivalent (modulo ℓ) to the original one. More particularly, for a, b ∈ S_{2}(F), one has [a, b] ∼^{ℓ} [b, a] if and only if 2 · {a, b} = 0 in K_{n}^{(ℓ)}F.

3.3 Lemma. For any a, b ∈ F^{×}, one has [a, b] ∼ [b^{−1}, ab^{2}] ∼ [a^{2}b, a^{−1}].

Proof: Note first that the product of both entries is the same for any of the three given sequences in S_{2}(F). It is further easy to check the equalities

{a, b} = {b^{−1}, ab^{2}} = {a^{2}b, a^{−1}}

in K_{2}F. This yields the statement. □

3.4 Corollary. If a = ±1, then [a, b] ∼ [b, a] for any b ∈ F^{×}.

Proof: Since a^{−1} = a, this is clear from the lemma. □

A generalisation of the last statement will be obtained in (3.9) below. Let S_{0}(F) denote the singleton set consisting of the empty sequence [ ].

Let further

S(F) = ⋃_{n∈ℕ}S_{n}(F) .

Hence (S(F), ⋄) is the free monoïd generated by F^{×}.

We are going to consider operations on sequences and how they behave with respect to chain equivalence. We say that a map

Φ : S(F) ⟶ S(F)

is compatible with chain ℓ-equivalence if it preserves lengths of sequences and if, for any n ∈ ℕ and any α, β ∈ S_{n}(F), the relation α ∼^{ℓ} β implies that Φ(α) ∼^{ℓ} Φ(β). We say that Φ as above is compatible if it is compatible with chain ℓ-equivalence for every ℓ ∈ ℕ. A compatible map Φ may or may not be an endomorphism of the monoïd S(F), and it does not need to be injective nor surjective.

Examples of compatible maps are given by rising the coefficients of a sequence to some power and by reversing the order of the coefficients.

Let α = [a_{1}, …, a_{n}] ∈ S_{n}(F). For any z ∈ ℤ, we write α^{z} for the sequence [a_{1}^{z}, …, a_{n}^{z}]. In particular, α^{−1} = [a_{1}^{−1}, …, a_{n}^{−1}]. Furthermore, we denote by α̃ the sequence with the entries of α in reversed order, i.e. α̃ = [a_{n}, …, a_{1}].

3.5 Proposition.

(a) For any z ∈ ℤ, the rule α ⟼ α^{z} defines a compatible endomorphism of S(F). In particular, α ⟼ α^{−1} is a compatible involution on S(F).

(b) The rule α ⟼ α̃ defines a compatible involution on S(F).

Proof: Only the compatibility (with respect to arbitrary ℓ) requires a proof. In view of the definitions, one needs to verify compatibility only for sequences α, β ∈ S_{n}(F) where n ≤ 2, thus to show for those that α ∼^{ℓ} β entails α^{z} ∼^{ℓ} β^{z} for any z ∈ ℤ, as well as α̃ ∼^{ℓ} β̃. For n = 1 this is obvious.

Let now α, β ∈ S_{2}(F) with α ∼^{ℓ} β. Using that, for any u, v ∈ F^{×} and any z ∈ ℤ, one has

{u^{z}v^{z}} = z · {uv} in K_{1}^{(ℓ)}F and {u^{z}, v^{z}} = z^{2} · {u, v} in K_{2}^{(ℓ)}F ,

one sees that α^{z} ∼^{ℓ} β^{z}, for any z ∈ ℤ. Similarily, using that {v, u} = − {u, v} in K_{2}^{(ℓ)}F for any u, v ∈ F^{×}, one sees that α̃ ∼^{ℓ} β̃. □

Given a sequence α = [a_{1}, …, a_{n}] ∈ S_{n}(F) and an element c ∈ F^{×}, we denote by c ∗ α the sequence [c^{ε_{1}}a_{1}, …, c^{ε_{n}}a_{n}], where ε_{i} = (−1)^{i} for i = 1, …, n. We call c ∗ α the conjugate of α by c. Note that conjugation is bijective, but it is in general not an endomorphism of the monoïd S(F). However, (−1) ∗ α = [−a_{1}, …, −a_{n}], so conjugation by −1 is an automorphism of S(F). As we shall see now, conjugation by an element of F^{×} is a compatible operation on S(F).

3.6 Proposition. Let α, β ∈ S_{n}(F), c ∈ F^{×}, and ℓ ∈ ℕ. Then α ∼^{ℓ} β if and only if c ∗ α ∼^{ℓ} c ∗ β.

Proof: By the definition of chain ℓ-equivalence, the statement needs to be proven only for n = 2. For any a, b, c ∈ F^{×}, we have {c^{−1}a, cb} = {c^{−1}, −ab} + {a, b} in K_{2}^{(ℓ)}F. Therefore, if a, b, a′, b′ ∈ F^{×} are such that

[a, b] ∼^{ℓ} [a′, b′], i.e. {ab} = {a′b′} in K_{1}^{(ℓ)}F and {a, b} = {a′, b′} in K_{2}^{(ℓ)}F,

then we obtain {c^{−1}a, cb} = {c^{−1}a′, cb′} and [c^{−1}a, cb] ∼^{ℓ} [c^{−1}a′, cb′]. This

proves one direction. Replacing c by c^{−1} then immediately yields the converse direction. □

For a sequence α = [a_{1}, …, a_{n}] in S_{n}(F), we denote by α^{∗} the sequence [−a_{n}^{−1}, …, −a_{1}^{−1}] in S_{n}(F). This defines an involution

∗ : S(F) ⟶ S(F) , α ⟼ α^{∗} .

3.7 Corollary. The involution ^{∗} : S(F) ⟶ S(F) is compatible.

Proof: The map ^{∗} is compatible, because it is the combination of three compatible operations on S(F) (which actually commute with each other), namely α ↦ α^{−1}, α ↦ α̃, and α ↦ (−1) ∗ α. □

For α ∈ S_{n}(F) and r ∈ ℕ we put r × α = α ⋄ … ⋄ α ∈ S_{rn}(F).

3.8 Proposition. For any α ∈ S_{n}(F) one has α ⋄ α^{∗} ∼ n × [1, −1] .

Proof: It follows from the definition of chain equivalence on S_{2}(F) that [a, −a^{−1}] ∼ [1, −1] holds for any a ∈ F^{×}. Since [1, −1] commutes up to chain equivalence with any element of S(F) by (3.4), the result follows. □

After having considered operations on S(F) which turned out to be compatible with chain equivalence, we may now ask which operations stabilise the equivalence classes of certain sequences.

3.9 Proposition. Let a_{1}, …, a_{n} ∈ F^{×} be such that a_{1} ⋯ a_{n} = ±1. Then

[a_{1}, …, a_{n}] ∼ [a_{n}^{−1}, a_{1}^{−1}, …, a_{n−1}^{−1}] .

Moreover, if n is odd, then

[a_{1}, …, a_{n}] ∼ [a_{1}^{−1}, …, a_{n}^{−1}] ∼ [a_{n}, a_{1}, …, a_{n−1}] .

Proof: Using n−1 times Lemma (3.3) we obtain

[a_{1}, …, a_{n}] ∼ [a_{1}, …, a_{n}a_{n−1}^{2}, a_{n−1}^{−1}]
∼ …
∼ [a_{n}(a_{1} ⋯ a_{n−1})^{2}, a_{1}^{−1}, …, a_{n−1}^{−1}] .

Since by hypothesis (a_{1} ⋯ a_{n−1})^{2} = a_{n}^{−2}, the first part of the statement follows. This also shows the second equivalence of the second part of the statement, which thus holds without condition on n.

Finally, if n is odd, then applying n times the first part yields

[a_{1}, …, a_{n}] ∼ [a_{1}^{−1}, …, a_{n}^{−1}] . □

3.10 Remark. Possibly the whole statement of the proposition is true for every n. The second part, which here is under the hypothesis that n is odd, can easily be proven for even n under the assumption that [a_{1}, …, a_{n}] is chain equivalent to a sequence having one entry equal to ±1.

3.11 Proposition. Let α = [a_{1}, …, a_{n}] ∈ S_{n}(F) with (a_{1} ⋯ a_{n}) = ±1. Then for any β ∈ S_{m}(F) one has α ⋄ β ∼ β ⋄ α^{ε}, where ε = (−1)^{m}.

Proof: We may assume that m = 1, so that β = [c] for some c ∈ F^{×}. Using Lemma (3.3) n times we obtain

α ⋄ β = [a_{1}, …, a_{n}, c] ∼ [a_{1}, …, a_{n−1}, ca_{n}^{2}, a_{n}^{−1}]
∼ …
∼ [c(a_{1} ⋯ a_{n})^{2}, a_{1}^{−1}, …, a_{n}^{−1}] = β ⋄ α^{−1} ,

because (a_{1} ⋯ a_{n})^{2} = 1, by the hypothesis. □

4 Virtual forms

We say that two sequences α, α′ ∈ S(F) are stably chain ℓ-equivalent if

α ⋄ (r × [1, −1]) ∼^{ℓ} α′ ⋄ (r × [1, −1])

holds for some r ∈ ℕ. We write M(F, ℓ) for the set of equivalence classes in S(F) modulo this equivalence relation. The elements of this set are called virtual forms of degree ℓ over F, or ℓ-forms for short. For α ∈ S(F) we denote by ⟨α⟩ the ℓ-form given by α. If α = [a_{1}, …, a_{n}] then we may write ⟨a_{1}, …, a_{n}⟩ instead of ⟨α⟩. The class of the empty sequence [ ] is also considered as an element of M(F, ℓ), and it is denoted by 0 and called the trivial form.

It is not clear whether stable chain ℓ-equivalence is really a coarser equivalence relation than chain ℓ-equivalence.

4.1 Question. For α, α′ ∈ S(F), does ⟨α⟩ = ⟨α′⟩ imply that α ∼^{ℓ} α′?

We define a (not necessarily commutative) operation + on M(F, ℓ). For α, β ∈ S(F), we set

⟨α⟩ + ⟨β⟩ = ⟨α ⋄ β⟩ .

Note that it follows from (3.4) that this operation is well-defined and associative. Therefore (M(F, ℓ), +) is a monoïd with neutral element 0. For n ∈ ℕ and φ ∈ M(F, ℓ), we denote by n × φ the n-fold sum φ + … + φ.

4.2 Lemma. The monoïd (M(F, ℓ), +) satisfies the cancellation law.

Proof: Given α, α′, β ∈ S(F) such that ⟨α⟩ + ⟨β⟩ = ⟨α′⟩ + ⟨β⟩, we claim that ⟨α⟩ = ⟨α′⟩. We may restrict to the case where β ∈ S_{1}(F), i.e. ⟨β⟩ = ⟨b⟩ for some b ∈ F^{×}. Note that ⟨b⟩ + ⟨−b^{−1}⟩ = ⟨b, −b^{−1}⟩ = ⟨1, −1⟩ by (3.8). Therefore ⟨α⟩ + ⟨β⟩ = ⟨α′⟩ + ⟨β⟩ implies that ⟨α⟩ + ⟨1, −1⟩ = ⟨α′⟩ + ⟨1, −1⟩. This means that α⋄[1, −1] and α′ ⋄[1, −1] are stably chain ℓ-equivalent. The same then holds for α and α′ so that ⟨α⟩ = ⟨α⟩. This shows that cancellation on the right is possible. The proof of cancellation on the left is analogous. □

We write ℍ for the ℓ-form ⟨1, −1⟩. Note that ⟨a, −a^{−1}⟩ = ℍ for any a ∈ F^{×} by (3.8). We denote by G(F, ℓ) the set of formal differences

φ − n × ℍ

where φ ∈ M(F, ℓ) and n ∈ ℕ. Since by (3.4) the ℓ-form n × ℍ lies in the center of M(F, ℓ) for any n ∈ ℕ, the operation + extends naturally to G(F, ℓ) and gives it the structure of a monoïd. Since M(F, ℓ) satisfies the cancellation law, the same is true for G(F, ℓ) and M(F, ℓ) can be seen as a submonoïd of G(F, ℓ) by identifying an ℓ-form φ with the difference φ − 0 × ℍ.

4.3 Proposition. G(F, ℓ) is a group.

Proof: We have to show that, for arbitrary n ∈ ℕ and α ∈ S(F), the difference ⟨α⟩ − n × ℍ has an inverse in G(F, ℓ). In fact, if m is the length of the sequence α, then the inverse is given by ⟨α^{∗}⟩ + (n − m) × ℍ, because ⟨α⟩ + ⟨α^{∗}⟩ = m × ℍ by (3.8). □

4.4 Example. Let us assume that char(F) ≠ 2 and consider the case ℓ = 2. By (3.1) and Witt’s Cancellation Theorem, 2-forms over F correspond bijectively to isometry classes of regular quadratic forms over F. In particular, Question (4.1) has a positive answer in this case and G(F, 2) is isomorphic to the Witt-Grothendieck group Ŵ(F).

We call G(F, ℓ) the Grothendieck group of ℓ-forms over F. Note that this group is not abelian unless 2 · K_{2}^{(ℓ)}F = 0 (e.g. for ℓ = 2).

4.5 Lemma. For any a, b ∈ F^{×}, we have ⟨a, b⟩ = ⟨b^{−1}, ab^{2}⟩ = ⟨a^{2}b, a^{−1}⟩.

Proof: This is immediate from (3.3). □

We next prove a universal property for the group G(F, ℓ).

4.6 Lemma. Let (G, ∘) be a group and g : F^{×} ⟶ G a map. Assume that the following hold for any a, b, a′, b′ ∈ F^{×}:

• if {a} = {b} in K_{1}^{(ℓ)}F, then g(a) = g(b) in G;

• if {ab} = {a′b′} in K_{1}^{(ℓ)}F and {a, b} = {a′, b′} in K_{2}^{(ℓ)}F, then g(a) ∘ g(b) = g(a′) ∘ g(b′) in G.

Then there is a unique group homomorphism Γ : G(F, ℓ) ⟶ G such that Γ(⟨a⟩) = g(a) for every a ∈ F^{×}.

Proof: We can define a homomorphism of monoïds S(F) ⟶ G by the rule [a_{1}, …, a_{n}] ⟼ g(a_{1}) ∘ … ∘ g(a_{n}). In view of the two conditions and since G is a group, it is obvious that the image of a sequence α ∈ S(F) under this homomorphism depends on α only up to stable chain ℓ-equivalence. This yields a homomorphism M(F, ℓ) ⟶ G which maps the ℓ-form ⟨a_{1}, …, a_{n}⟩ to g(a_{1})∘…∘g(a_{n}). Since the group G(F, ℓ) is generated by M(F, ℓ) and since G is a group, it is clear that this map extends to a map Γ : G(F, ℓ) ⟶ G with the desired property. The uniqueness of Γ is obvious. □

4.7 Remark. The lemma shows that (G(F, ℓ), +) is equal to the group defined by generators and relations in the following way. G(F, ℓ) is generated by elements ⟨a⟩ with a ∈ F^{×}, and the defining relations are:

(V1)  ⟨a⟩ = ⟨b⟩  if  {a} = {b}  in K_{1}^{(ℓ)}F ;

(V2) ⟨a⟩ + ⟨b⟩ = ⟨a′⟩ + ⟨b′⟩ if \{ \begin{matrix}\{ab\} = \{a′b′\} & in K_{1}^{(ℓ)}F and \\ \{a , b\} = \{a′ , b′\} & in K_{2}^{(ℓ)}F . \end{matrix}

We will frequently use these rules without particular mention.

Let φ be an ℓ-form over F, say φ = ⟨a_{1}, …, a_{r}⟩. We shall refer to r as the rank of φ and denote it by rk(φ). We further put d_{1}(φ) = {a_{1} ⋯ a_{r}} ∈ K_{1}^{(ℓ)}F and call this the determinant of φ. This yields two group homomorphisms

rk : G(F, ℓ) ⟶ ℤ ,
d_{1} : G(F, ℓ) ⟶ K_{1}^{(ℓ)}F .

The scrupulous reader may apply (4.6) to check that these homomorphisms are actually well-defined. Note that the group G(F, ℓ) is generated by the ℓ-forms of rank one.

For an ℓ-form φ given by φ = ⟨α⟩ with α ∈ S(F), we use the notations φ^{∗} = ⟨α^{∗}⟩, φ^{z} = ⟨α^{z}⟩ for z ∈ ℤ, in particular φ^{−1} = ⟨α^{−1}⟩, and further c ∗ φ = ⟨c ∗ α⟩ for c ∈ F^{×}. By (3.5), (3.6), and (3.7), these operations on ℓ-forms are well-defined.

4.8 Proposition. For any ℓ-form φ over F, we have:

(a) φ + φ^{∗} = rk(φ) × ℍ;

(b) (c ∗ φ)^{∗} = \{ \begin{matrix}c ∗ φ^{∗} & if rk(φ) is even, \\ (c^{−1}) ∗ φ^{∗} & if rk(φ) is odd, \end{matrix} for any c ∈ F^{×}.

Proof: Part (a) follows from (3.8), and (b) is easily checked. □

We may extend the conjugation operation to elements of G(F, ℓ): for a ∈ F^{×}, φ ∈ M(F, ℓ), and m ∈ ℕ, we put

c ∗ (φ − m × ℍ) = c ∗ φ − m × ℍ .

We denote the inverse of an element ξ ∈ G(F, ℓ) by −ξ. (This should not be confused with the element (−1) ∗ ξ.) If ξ, ζ ∈ G(F, ℓ), then ξ − ζ is meant to be ξ + (−ζ). Furthermore, with ξ ∈ G(F, ℓ) and n ∈ ℕ, we write n × ξ for the n-fold sum ξ + ⋯ + ξ.

4.9 Proposition. Let φ be an ℓ-form over F with d_{1}(φ)=0 or d_{1}(φ)={−1}. Then φ = φ^{−1}. Moreover, if a_{1}, …, a_{n} ∈ F^{×} are such that φ = ⟨a_{1}, …, a_{n}⟩, then also φ = ⟨a_{n}, a_{1}, …, a_{n−1}⟩.

Proof: Let a_{1}, …, a_{n} ∈ F^{×} be such that φ = ⟨a_{1}, …, a_{n}⟩. The hypothesis on d_{1}(φ) says that a_{1} ⋯ a_{n} = ±c for some c ∈ F^{×ℓ}. Since ℓ-forms remain unchanged if one entry of a representing sequence is multiplied by an element of F^{×ℓ}, we may assume that c = 1, whence a_{1} ⋯ a_{n} = ±1. If n is odd, then the claims now follow directly from (3.9). If n is even, then we apply (3.9) to φ ⊥ ⟨1⟩ instead and use that ⟨1⟩ lies in the center of G(F, ℓ). □

4.10 Corollary. The kernel of d_{1} : G(F, ℓ) ⟶ K_{1}^{(ℓ)}F is contained in the center of G(F, ℓ).

Proof: Let φ and ψ be ℓ-forms over F with d_{1}(φ) = 0. Then φ + ψ = ψ + φ^{ε} with ε = ±1 by (3.11) and φ^{ε} = φ by (4.9). Hence φ and ψ commute. □

4.11 Corollary. Let φ be an ℓ-form of even rank and trivial determinant over F. Then c^{2} ∗ φ = φ for any c ∈ F^{×}.

Proof: Applying (4.9) to φ and to c ∗ φ yields

c ∗ φ = (c ∗ φ)^{−1} = c^{−1} ∗ φ^{−1} = c^{−1} ∗ φ ,

whence c^{2} ∗ φ = φ. □

For any ℓ-form φ over F, we define its companion form φ^{∘} to be

φ^{∘} = \{ \begin{matrix}m × ℍ + ⟨c⟩ & if rk(φ) = 2m + 1, \\ m × ℍ + ⟨1, c⟩ & if rk(φ) = 2m + 2, \end{matrix}

where c ∈ F^{×} is chosen such that d_{1}(φ) = d_{1}(φ^{∘}).

4.12 Proposition. Let ξ ∈ G(F, ℓ). To have ξ = φ − φ^{∘} for some ℓ-form φ over F, it is necessary and sufficient that rk(ξ) = 0 and d_{1}(ξ) = 0.

Proof: We write ξ = φ − m × ℍ with φ ∈ M(F, ℓ) and m ∈ ℕ. If rk(ξ) = 0 and d_{1}(ξ) = 0, then rk(φ) = 2m and d_{1}(φ) = d_{1}(m × ℍ) = {(−1)^{m}}, so that φ^{∘} = m × ℍ and ξ = φ − φ^{∘}. Conversely, for any ℓ-form φ over F one has rk(φ − φ^{∘}) = 0 and d_{1}(φ − φ^{∘}) = d_{1}(φ) − d_{1}(φ^{∘}) = 0. □

Given x ∈ ℝ, let [x] denote the integral part of x, that is, the unique integer such that [x] ≤ x < [x] + 1.

4.13 Proposition. Let φ be an ℓ-form over F and let m = [\frac{1}{2}(rk(φ) − 1)]. There exist ℓ-forms ϑ_{1}, …, ϑ_{m} over F each of rank 4 and trivial determinant such that

φ − φ^{∘} = (ϑ_{1} − 2 × ℍ) + ⋯ + (ϑ_{m} − 2 × ℍ) .

Proof: We proceed by induction on m. If m = 0, then φ = φ^{∘} and thus the statement holds trivially. Let now m > 0. We may write φ = φ′ + ⟨x, y, z⟩ for certain x, y, z ∈ F^{×} and an ℓ-form φ′. We put ϑ = ⟨x, y, z, (xyz)^{−1}⟩ and ψ = φ′ + ⟨−xyz⟩. Since d_{1}(ϑ) = 0, we obtain φ + ℍ = φ′ + ϑ + ⟨−xyz⟩ = ψ + ϑ by (4.10). Since rk(ψ) = rk(φ) − 2 and d_{1}(φ) = d_{1}(ψ + ℍ), we have φ^{∘} = ψ^{∘} + ℍ. Thus using (4.10) yields

φ − φ^{∘} = ψ − ψ^{∘} + ϑ − 2 × ℍ .

Now we apply the induction hypothesis to ψ. □

5 Pfister forms

For a ∈ F^{×}, we put ⟨⟨a⟩⟩ = ⟨1, −a⟩ and call this a 1-fold Pfister form (of degree ℓ). Given n > 1 and a_{1}, …, a_{n} ∈ F^{×}, we define recursively

⟨⟨a_{1}, …, a_{n}⟩⟩ = φ + (a_{1} ∗ φ)^{∗} = φ + a_{1} ∗ φ^{∗} where φ = ⟨⟨a_{2}, …, a_{n}⟩⟩ ,

and we call ⟨⟨a_{1}, …, a_{n}⟩⟩ an n-fold Pfister form. This is an ℓ-form of rank 2^{n}. Note that, for a ∈ F^{×}, we have ⟨⟨a⟩⟩ = ⟨1⟩ + (a ∗ ⟨1⟩)^{∗}.

5.1 Lemma. For any a, b ∈ F^{×}, we have ⟨⟨a, b⟩⟩ = ⟨1, −a, −b, (ab)^{−1}⟩.

Proof: We compute ⟨⟨a, b⟩⟩ = ⟨1, −b⟩ + ⟨a^{−1}, −ab⟩^{∗} = ⟨1, −b, (ab)^{−1}, −a⟩ and then use (4.9). □

Let a, b, c, d ∈ F^{×}. In order to have that the ℓ-forms ⟨⟨a, b⟩⟩ and ⟨⟨c, d⟩⟩ are equal, it is necessary to have {a, b} = {c, d} in K_{2}^{(ℓ)}F. This will become obvious from (7.4), below. On the other hand, if one has {ab} = {cd} in K_{1}^{(ℓ)}F and {a, b} = {c, d} in K_{2}^{(ℓ)}F, then it follows easily using (5.1) that ⟨⟨a, b⟩⟩ = ⟨⟨c, d⟩⟩.

5.2 Proposition. Assume that a, b, c ∈ F^{×} are such that {a, b} = {b, c} in K_{2}^{(ℓ)}F. Then ⟨⟨a, b⟩⟩ = ⟨⟨b, c⟩⟩.

Proof: Let a, b, c ∈ F^{×} with {a, b} = {b, c}. By (4.9) and (4.5) we have ⟨−a, −b, (ab)^{−1}⟩ = ⟨−b, (ab)^{−1}, −a⟩ = ⟨−b, −a^{−1}, ab^{−1}⟩. With (5.1) we conclude that ⟨⟨a, b⟩⟩ = ⟨⟨b, a^{−1}⟩⟩.

Using now that {a, b} = {b, c} = −{c, b} we obtain

{−a^{−1}, b^{−1}a} = {−a, b} = − {−c, b} = {−c, b^{−1}} = {−c, (bc)^{−1}} .

Therefore ⟨−a^{−1}, b^{−1}a⟩ = ⟨−c, (bc)^{−1}⟩ and thus ⟨⟨b, a^{−1}⟩⟩ = ⟨⟨b, c⟩⟩. □

5.3 Corollary. If a, b ∈ F^{×} are such that {a, b} = 0, then ⟨⟨a, b⟩⟩ = 2 × ℍ.

Proof: This is follows from (5.2), because {b, 1} = {1, 1} = 0. □

5.4 Question. Assume that a, b, c, d ∈ F^{×} are such that {a, b} = {c, d} in K_{2}^{(ℓ)}F. Does it follow that the ℓ-forms ⟨⟨a, b⟩⟩ and ⟨⟨c, d⟩⟩ coincide?

In the case where ℓ = 2 ≠ char(F), a positive answer to this question is contained in [3, Theorem 1.8.]). We will see in (7.13) that the answer is positive in several other cases, in particular for any odd ℓ.

5.5 Problem. Let a, b, c, d ∈ F^{×} be such that {a, b} = {c, d} in K_{2}^{(ℓ)}F. It is not known whether in this situation one can always find a chain of elements a_{1}, …, a_{n} ∈ F^{×} with a_{1} = a, a_{2} = b, a_{n−1} = c, a_{n} = d and such that {a_{i−1}, a_{i}} = {a_{i}, a_{i+1}} for 1 < i < n. For ℓ = 2 and char(F) ≠ 2 this is possible and one can achieve this with n = 5 ([6, Chap. V, §4]). If ℓ = 3 and if F contains a primitive 3rd root of unity (in particular char(F) ≠ 3), then an analogous statement where n = 7 follows from [9, Corollary 2.2.].

6 Subgroups

We are going to define a sequence of subgroups (G^{n}(F, ℓ))_{n∈ℕ} of G(F, ℓ) generalizing the powers of the fundamental ideal in the Witt-Grothendieck ring of quadratic forms.

We put G^{0}(F, ℓ) = G(F, ℓ). For n > 0, we define G^{n}(F, ℓ) to be the group generated by the differences ξ − c ∗ ξ where ξ ∈ G^{n−1}(F, ℓ) and c ∈ F^{×}. Clearly G^{n}(F, ℓ) is a subgroup of G^{n−1}(F, ℓ).

6.1 Proposition. G^{1}(F, ℓ) is equal to the kernel of rk : G(F, ℓ) ⟶ ℤ.

Proof: It is clear that every element of G^{1}(F, ℓ) has trivial rank. On the other hand, the kernel of rk is generated by the differences ⟨a⟩ − ⟨1⟩ = ⟨a⟩ − a ∗ ⟨a⟩ with a ∈ F^{×}, and these belong to G^{1}(F, ℓ). □

6.2 Proposition. The groups G^{n}(F, ℓ) with n ≥ 2 lie in the center of G(F, ℓ).

Proof: For ξ ∈ G^{1}(F, ℓ) and c ∈ F^{×}, we have rk(ξ) = 0, thus d_{1}(c∗ξ) = d_{1}(ξ) and d_{1}(ξ − c ∗ ξ) = 0. By (4.10) the kernel of d_{1} : G(F, ℓ) ⟶ K_{1}^{(ℓ)}F is contained in the center of G(F, ℓ), so the statement follows. □

6.3 Lemma. For any a, b, c ∈ F^{×}, one has

⟨a, b, c, (abc)^{−1}⟩ = ⟨⟨−a, −b⟩⟩ + ⟨⟨−ab, −c⟩⟩ − ⟨⟨−1, −ab⟩⟩ .

Proof: We compute

⟨a, b, c, (abc)^{−1}⟩ + ⟨⟨−1, −ab⟩⟩ = ⟨1, a, b, (ab)^{−1}⟩ + ⟨1, ab, c, (abc)^{−1}⟩
= ⟨⟨−a, −b⟩⟩ + ⟨⟨−ab, −c⟩⟩
□

6.4 Theorem. G^{2}(F, ℓ) is equal to the kernel of d_{1} : G^{1}(F, ℓ) ⟶ K_{1}^{(ℓ)}F. Furthermore, G^{2}(F, ℓ) consists of all the differences φ−φ^{∘} with φ ∈ M(F, ℓ), and it is generated by the differences ⟨⟨a, b⟩⟩ − 2 × ℍ with a, b ∈ F^{×}.

Proof: We denote by H the kernel of d_{1} : G^{1}(F, ℓ) ⟶ K_{1}^{(ℓ)}F. By (4.12), it consists of the differences φ−φ^{∘} with φ ∈ M(F, ℓ). By (4.13), H is generated by the differences ϑ − 2 × ℍ where ϑ is an ℓ-form over F of rank 4 and trivial determinant. Note that any such form can be written as ϑ = ⟨x, y, z, (xyz)^{−1}⟩ with x, y, z ∈ F^{×}, and then

ϑ−2×ℍ = (⟨⟨−x, −y⟩⟩−2×ℍ)+(⟨⟨−xy, −z⟩⟩−2×ℍ)−(⟨⟨−1, −xy⟩⟩−2×ℍ) ,

by (6.3). Therefore H is already generated by the differences ⟨⟨a, b⟩⟩ − 2 × ℍ with a, b ∈ F^{×}. Since ⟨⟨a, b⟩⟩ − 2 × ℍ = ⟨1, −b⟩ − a ∗ ⟨1, −b⟩, it follows that H is contained in G^{2}(F, ℓ). On the other hand, it is clear that G^{2}(F, ℓ) is contained in H, the kernel of d_{1} : G^{1}(F, ℓ) ⟶ K_{1}^{(ℓ)}F. □

6.5 Corollary. For any n ≥ 1, the group G^{n}(F, ℓ) is generated by the differences π − 2^{n−1} × ℍ where π is an n-fold Pfister form.

Proof: For n = 1 this is easy to see and for n = 2 the statement is contained in (6.4). We proceed by induction on n. Let n > 2. By definition, G^{n}(F, ℓ) is generated by the elements ξ − c ∗ ξ with ξ ∈ G^{n−1}(F, ℓ) and c ∈ F^{×}. We want to show that ξ − c ∗ ξ is a sum of elements of the form π − 2^{n−1} × ℍ where π is an n-fold Pfister form. Applying the induction hypothesis to ξ and using that G^{n−1}(F, ℓ) is commutative by (6.2), we restrict to the case where ξ = ρ − 2^{n−2} × ℍ for some (n−1)-fold Pfister form ρ. We obtain

ξ − c ∗ ξ = ρ − c ∗ ρ = ρ + c ∗ ρ^{∗} − 2^{n−1} × ℍ ,

and since ρ + c ∗ ρ^{∗} is an n-fold Pfister form, this finishes the proof. □

6.6 Question. Do we have ⋂_{n=0}^{∞} G^{n}(F, ℓ) = 0?

In the case where ℓ = 2 ≠ char(F), the Arason-Pfister Hauptsatz gives a positive answer to this question (cf. [6, Chap. X, §5]). There are other cases where the answer is positive, in fact by rather simple reasons.

6.7 Theorem. Assume that ℓ is odd or that F^{×} = F^{×2}. Then G^{3}(F, ℓ) = 0.

Proof: By hypothesis, F^{×}/F^{×ℓ} is 2-divisible. Therefore G^{3}(F, ℓ) is generated by elements ξ − c^{2} ∗ ξ with c ∈ F^{×} and ξ ∈ G^{2}(F, ℓ). Now, for ξ ∈ G^{2}(F, ℓ) there is φ ∈ M(F, ℓ) and m ∈ ℕ such that d_{1}(φ) = 0, rk(φ) = 2m, and ξ = φ − m × ℍ. Using (4.11), we obtain for any c ∈ F^{×} that

ξ − c^{2} ∗ ξ = φ − c^{2} ∗ φ = 0 ,

and this finishes the proof. □

6.8 Proposition. The commutator subgroup of G(F, ℓ) is generated by the elements of the shape ⟨⟨a^{2}, b⟩⟩ − ⟨⟨1, 1⟩⟩ with a, b ∈ F^{×}.

Proof: Let ξ, ζ ∈ G(F, ℓ). We want to compute the commutator of ξ and ζ. We choose a, b ∈ F^{×} such that d_{1}(ξ) = {a} and d_{1}(ζ) = {b} in K_{1}^{(ℓ)}F. It follows that ξ ≡ ⟨a⟩ and ζ ≡ ⟨b⟩ modulo the center of G(F, ℓ). This yields

ξ + ζ − ξ − ζ = ⟨a⟩ + ⟨b⟩ − ⟨a⟩ − ⟨b⟩ = ⟨a, b, −a^{−1}, −b^{−1}⟩ − ⟨⟨1, 1⟩⟩ .

Furthermore, we have

⟨a, b, −a^{−1}, −b^{−1}⟩ = ⟨a, −a, a^{−2}b, −b^{−1}⟩ = ⟨1, −a^{2}, a^{−2}b, −b^{−1}⟩ = ⟨⟨a^{2}, b⟩⟩.

Therefore ξ + ζ − ξ − ζ = ⟨⟨a^{2}, b⟩⟩ − ⟨⟨1, 1⟩⟩, and the statement follows. □

6.9 Corollary. If ℓ is odd, then G^{2}(F, ℓ) is the commutator subgroup of G(F, ℓ).

Proof: If ℓ is odd, then the generators of G^{2}(F, ℓ) given by (6.5) can all be written as ⟨⟨1, 1⟩⟩−⟨⟨a^{2}, b⟩⟩ (for given c, b ∈ F^{×}, one may set a = c^{\frac{ℓ+1}{2}} in order to have ⟨⟨c, b⟩⟩ = ⟨⟨a^{2}, b⟩⟩). □

6.10 Theorem. If −1 ∈ F^{×ℓ}, then ℓ × ξ = 0 for any ξ ∈ G^{1}(F, ℓ).

Proof: Assuming that −1 ∈ F^{×ℓ}, we have ⟨−x⟩ = ⟨x⟩ for any x ∈ F^{×}.

Let d ∈ F^{×}. Induction on i shows that the equalities ⟨d^{i−1}, d⟩ = ⟨d^{i}, 1⟩ and i × (⟨d⟩ − ⟨1⟩) = ⟨d^{i}⟩ − ⟨1⟩ hold for any i ≥ 1. In particular, for ξ = ⟨d⟩ − ⟨1⟩ we have that ℓ × ξ = 0.

Let a, b ∈ F^{×} and i ≥ 1. Using (5.2) and (4.10), we compute

⟨⟨a^{i}, b⟩⟩ + ⟨⟨a, b⟩⟩ = ⟨⟨b, (a^{i}b)^{−1}⟩⟩ + ⟨⟨a, b⟩⟩
= ⟨1, 1, b, a^{−i}b^{−1}, a^{i}, a, b, (ab)^{−1}⟩
= ⟨1, 1, b, a^{−i}b^{−1}, a^{i+1}, 1, b, (ab)^{−1}⟩
= ⟨b, a^{−i}b^{−1}, 1, a^{i+1}, b, 1, 1, (ab)^{−1}⟩
= ⟨b, a^{−i}b^{−1}, 1, a^{i+1}, b, (a^{i+1}b)^{−1}, a^{i+1}b, (ab)^{−1}⟩
= ⟨b, a^{−i}b^{−1}⟩ + ⟨⟨a^{i+1}, b⟩⟩ + ⟨a^{i+1}b, (ab)^{−1}⟩
= ⟨⟨a^{i+1}, b⟩⟩ + ⟨b, a^{−i}b^{−1}, a^{i+1}b, (ab)^{−1}⟩ .

Since {a^{−i}b^{−1}, a^{i+1}b} = {a^{−i}b^{−1}, a} = {a, b} = {b^{−1}, ab} in K_{2}^{(ℓ)}F, we have further that ⟨a^{−i}b^{−1}, a^{i+1}b⟩ = ⟨b^{−1}, ab⟩. Therefore we obtain

⟨⟨a^{i}, b⟩⟩ + ⟨⟨a, b⟩⟩ = ⟨⟨a^{i+1}b⟩⟩ + ⟨b, b^{−1}, ab, (ab)^{−1}⟩ = ⟨⟨a^{i+1}, b⟩⟩ + ⟨⟨1, 1⟩⟩.

By induction on i we deduce that

i × (⟨⟨a, b⟩⟩ − ⟨⟨1, 1⟩⟩) = ⟨⟨a^{i}, b⟩⟩ − ⟨⟨1, 1⟩⟩.

Since ⟨⟨a^{ℓ}, b⟩⟩ = ⟨⟨1, 1⟩⟩, we have in particular for ξ = (⟨⟨a, b⟩⟩ − ⟨⟨1, 1⟩⟩) that ℓ × ξ = 0.

Since G^{2}(F, ℓ) is an abelian group, generated by elements (⟨⟨a, b⟩⟩−⟨⟨1, 1⟩⟩) with a, b ∈ F^{×}, the previous implies that ℓ×ξ = 0 holds for any ξ ∈ G^{2}(F, ℓ).

Let now ξ ∈ G^{1}(F, ℓ) be arbitrary. With d ∈ F^{×} such that d_{1}(ξ) = {d} in K_{1}^{(ℓ)}F and ζ = ξ − ⟨d⟩ + ⟨1⟩ we have ζ ∈ G^{2}(F, ℓ). By what we have shown, ℓ × ζ = 0 and ℓ × (⟨d⟩ − ⟨1⟩) = 0. Since ζ lies in the center of G(F, ℓ), it follows that ℓ × ξ = ℓ × (ζ + ⟨d⟩ − ⟨1⟩) = 0. □

Let R^{(ℓ)}F denote the subgroup of F^{×} consisting of those x ∈ F^{×} satisfying {x, y} = 0 in K_{2}^{(ℓ)}F for any y ∈ F^{×}. We call R^{(ℓ)}F the ℓ-radical of F. Note that F^{×ℓ} ⊂ R^{(ℓ)}F. Let R̄^{(ℓ)}F denote the corresponding subgroup of K_{1}^{(ℓ)}F, consisting of the symbols {r} with r ∈ R^{(ℓ)}F. In the case where char(F) ≠ 2, the group R^{(2)}F was introduced by Kaplansky in [5] and called ‘the radical of F ’ and denoted by R(F). 6.11 Proposition. If ℓ is odd, then the center of G(F, ℓ) is equal to the preimage of R̄^{(ℓ)}F under d_{1} : G(F, ℓ) ⟶ K_{1}^{(ℓ)}F. Proof: Let ξ ∈ G(F, ℓ). Let d ∈ F^{×} be such that d_{1}(ξ) = {d}. Then ξ ≡ ⟨d⟩ modulo the center of G(F, ℓ). Thus ξ is in the center of G(F, ℓ) if and only if the identity ⟨d, e⟩ = ⟨e, d⟩ holds for every e ∈ F^{×}. But for any e ∈ F^{×} we have the equivalences

⟨d, e⟩ = ⟨e, d⟩ ⟺ {d, e} = {e, d} ⟺ 2 · {d, e} = 0 ⟺ {d, e} = 0 .

Therefore ξ belongs to the center of G(F, ℓ) if and only if d ∈ R^{(ℓ)}F. □

7 Delzant classes

Stiefel-Whitney classes of quadratic forms were defined by Delzant in [2] and later addapted by Milnor in [8] in such way that these maps take their values in K-groups instead of Galois cohomology groups. Here they will be generalised to what shall be named Delzant classes. They are disigned as a tool to perform computations in Milnor K-theory.

We denote by U(F, ℓ) the multiplicative group of the ring K̂_{∗}^{(ℓ)}F and by U^{1}(F, ℓ) the subgroup consisting of the elements with constant term equal to 1 (in K_{0}^{(ℓ)}F = ℤ/ℓℤ). By (4.6), there is a unique group homomorphism

d : G(F, ℓ) ⟶ U^{1}(F, ℓ)

which maps ⟨x⟩ to 1 + {x}, for any x ∈ F^{×}. We call d the Delzant homomorphism (modulo ℓ).

For any n ∈ ℕ let d_{n} : G(F, ℓ) ⟶ K_{n}^{(ℓ)}F be the composition of d with the projection from K̂_{∗}^{(ℓ)}F to the nth component K_{n}^{(ℓ)}F. In other terms, for ξ ∈ G(F, ℓ) we write

d(ξ) = ∑_{n=0}^{∞} d_{n}(ξ)

with d_{n}(ξ) ∈ K_{n}^{(ℓ)}F (n ∈ ℕ); we call d_{n}(ξ) the n-th Delzant class of ξ. Note that d_{0} : G(F, ℓ) ⟶ K_{0}^{(ℓ)}F is the constant map 1 and d_{1} : G(F, ℓ) ⟶ K_{1}^{(ℓ)}F is the determinant homomorphism defined earlier.

The interest of these maps in the study of K-groups lies in the fact that K_{n}^{(ℓ)}F is generated by the image of d_{n} : G(F, ℓ) ⟶ K_{n}^{(ℓ)}F. Indeed K_{n}^{(ℓ)}F (n ≥ 1) is generated by symbols and any symbol {a_{1}, …, a_{n}} is equal to the nth Delzant class of the ℓ-form ⟨a_{1}, …, a_{n}⟩.

7.1 Proposition. For any φ ∈ M(F, ℓ) and n > rk(φ), one has d_{n}(φ) = 0.

Proof: Let φ be an ℓ-form of rank r over F. With a_{1}, …, a_{r} ∈ F^{×} such that φ = ⟨a_{1}, …, a_{r}⟩, we see that d(φ) = (1 + {a_{1}}) ⋯ (1 + {a_{r}}) lies in ⨁_{i=0}^{r} K_{i}^{(ℓ)}F, whence d_{n}(φ) = 0 for n > r. □

7.2 Proposition. For any ξ_{1}, …, ξ_{k} ∈ G(F, ℓ) one has

d_{n}(ξ_{1} + ⋯ + ξ_{k}) = ∑_{i_{1}+···+i_{k}=n^{i_{1},...,i_{k}≥0}}(d_{i_{1}}(ξ_{1}) ⋯ d_{i_{k}}(ξ_{k})) .

Proof: This is straightforward from the definitions of d and d_{n} (n ≥ 1). □

7.3 Lemma. Let a, b ∈ F^{×}. Then d (⟨⟨1, 1⟩⟩−⟨⟨a, b⟩⟩) = 1 − {a, b}.

Proof: One easily checks that d(⟨⟨a, b⟩⟩) = 1 + {a, b} + {−1, −1} and d(⟨⟨1, 1⟩⟩) = 1 + {−1, −1} = (1 − {a, b}) · d(⟨⟨a, b⟩⟩). The statement now follows because d : G(F, ℓ) ⟶ U^{1}(F, ℓ) is a group homomorphism. □

7.4 Proposition. The restricted map d_{2} : G^{2}(F, ℓ) ⟶ K_{2}^{(ℓ)}F is a surjective homomorphism. For any a, b ∈ F^{×}, it maps ⟨⟨a, b⟩⟩ − ⟨⟨1, 1⟩⟩ to {a, b}.

Proof: For any ξ, ζ ∈ G(F, ℓ) we obtain from (7.2) that

d_{2}(ξ + ζ) = d_{2}(ξ) + d_{1}(ξ)d_{1}(ζ) + d_{2}(ζ) .

The term d_{1}(ξ)d_{1}(ζ) is zero in case one of ξ and ζ belongs to G^{2}(F, ℓ). Therefore d_{2} : G^{2}(F, ℓ) ⟶ K_{2}^{(ℓ)}F is a homomorphism. It follows from (7.3) that d_{2} (⟨⟨a, b⟩⟩ − ⟨⟨1, 1⟩⟩) = {a, b} for any a, b ∈ F^{×}. Using (6.5), the surjectivity is now obvious. □

7.5 Corollary. One has G^{2}(F, ℓ) = 0 if and only if K_{2}^{(ℓ)}F = 0, and in this case G(F, ℓ) is commutative and G^{1}(F, ℓ) is isomorphic to F^{×}/F^{×ℓ}.

Proof: If G^{2}(F, ℓ) = 0, then K_{2}^{(ℓ)}F = 0 by (7.4). Conversely, assume that K_{2}^{(ℓ)}F vanishes. Then (5.3) and (6.4) together imply that G^{2}(F, ℓ) = 0. In this case d_{1} : G^{1}(F, ℓ) ⟶ K_{1}^{(ℓ)}F is an isomorphism, whence G^{1}(F, ℓ) ≅ K_{1}^{(ℓ)}F ≅ F^{×}/F^{×ℓ}, and using (6.8) it follows that G(F, ℓ) is commutative. □

The next aim is to show that G^{3}(F, ℓ) is equal to the kernel of the homomorphism d_{2} : G^{2}(F, ℓ) ⟶ K_{2}^{(ℓ)}F.

7.6 Lemma. Let a, b, c ∈ F^{×}. Then

⟨⟨a, b, c⟩⟩ = ⟨⟨a, c⟩⟩ + ⟨⟨b, c⟩⟩ − ⟨⟨ab, c⟩⟩ + 2 × ℍ .

Proof: Since ⟨⟨a, b, c⟩⟩ = ⟨⟨b, c⟩⟩ + a ∗ ⟨⟨b, c⟩⟩^{∗}, the statement follows from

a ∗ ⟨⟨b, c⟩⟩^{∗} = ⟨−a^{−1}bc, ac^{−1}, (ab)^{−1}, −a⟩
=^{(4.5)} ⟨a^{−1}c, −(ab)c^{−1}, (ab)^{−1}, −a⟩
=^{(4.9)} ⟨−a, a^{−1}c, −(ab)c^{−1}, (ab)^{−1}⟩
= ⟨−a, a^{−1}c⟩ + ⟨−c^{−1}, c⟩ + ⟨−(ab)c^{−1}, (ab)^{−1}⟩ − ℍ
= ⟨1, −a, a^{−1}c, −c^{−1}⟩ + ⟨c, −(ab)c^{−1}, (ab)^{−1}, −1⟩ − 2 × ℍ
= ⟨⟨c^{−1}, a⟩⟩ + ⟨⟨c^{−1}, ab⟩⟩^{∗} − 2 × ℍ
= ⟨⟨a, c⟩⟩ − ⟨⟨c^{−1}, ab⟩⟩ + 2 × ℍ
= ⟨⟨a, c⟩⟩ − ⟨⟨ab, c⟩⟩ + 2 × ℍ .
□

7.7 Corollary. For any x, y, z ∈ F^{×}, one has the following congruences modulo G^{3}(F, ℓ):

⟨⟨xy, z⟩⟩ + 2 × ℍ ≡ ⟨⟨x, z⟩⟩ + ⟨⟨y, z⟩⟩  (1)
⟨⟨x, yz⟩⟩ + 2 × ℍ ≡ ⟨⟨x, y⟩⟩ + ⟨⟨x, z⟩⟩  (2)

Proof: The lemma immediately yields (1). To obtain (2), one uses that ⟨⟨x, yz⟩⟩ = ⟨⟨yz, x^{−1}⟩⟩ by (5.2). □

7.8 Lemma. For any a, b, c ∈ F^{×} and n ≥ 1 one has

d_{n} (⟨⟨a, b, c⟩⟩ − 4 × ℍ) = \{ \begin{matrix}\{−1\}^{n−3} · \{a, b, c\} & if 4 | n, \\ 0 & otherwise. \end{matrix}

Proof: For any n ≥ 1, let s_{n} denote the term on the right hand side in the claimed equality. It is easily checked that

(1 + ∑_{n=1}^{∞} s_{n}) · (1 − {ab, c} + {−1, a, b, c}) = 1 − {ab, c} .

Using (7.6) and (7.3) we obtain that

d (⟨⟨a, b, c⟩⟩ − 4 × ℍ) = (1 − {a, c})^{−1} · (1 − {b, c})^{−1} · (1 − {ab, c})
= (1 − {ab, c} + {−1, a, b, c})^{−1} · (1 − {ab, c}) .

Therefore d (⟨⟨a, b, c⟩⟩ − 4 × ℍ) = (1 + ∑_{n=1}^{∞} s_{n}). □

7.9 Theorem. G^{3}(F, ℓ) is the kernel of d_{2} : G^{2}(F, ℓ) ⟶ K_{2}^{(ℓ)}F.

Proof: By (7.4), the restriction of d_{2} to G^{2}(F, ℓ) is a homomorphism. It follows from (6.5) and (7.8) that d_{2} is trivial on G^{3}(F, ℓ). Hence we obtain a homomorphism \overline{d}_{2} : G^{2}(F, ℓ)/G^{3}(F, ℓ) ⟶ K_{2}^{(ℓ)}F which maps the difference ⟨⟨a, b⟩⟩ − ⟨⟨1, 1⟩⟩ to the symbol {a, b}.

Now we consider the pairing F^{×} ×F^{×} ⟶ G^{2}(F, ℓ)/G^{3}(F, ℓ) which associates to a pair (a, b) the class of ⟨⟨a, b⟩⟩−⟨⟨1, 1⟩⟩. Since ⟨⟨a, b⟩⟩ = ⟨⟨1, 1⟩⟩ whenever {a, b} = 0, this induces a homomorphism K_{2}^{(ℓ)}F ⟶ G^{2}(F, ℓ)/G^{3}(F, ℓ) which is inverse to \overline{d}_{2}. Thus \overline{d}_{2} is an isomorphism. □

7.10 Remark. From (7.9) we readily obtain an exact sequence

0 ⟶ K_{2}^{(ℓ)}F ⟶^{(\overline{d_{2}})^{−1}} G^{1}(F, ℓ)/G^{3}(F, ℓ) ⟶^{\overline{d_{1}}} K_{1}^{(ℓ)}F ⟶ 0 .

This turns out to be the group extension constructed from the 2-cocycle K_{1}^{(ℓ)}F × K_{1}^{(ℓ)}F ⟶ K_{2}^{(ℓ)}F given by the multiplication in K_{∗}^{(ℓ)}F.

7.11 Corollary. If −1 ∈ F^{×}^{ℓ}, then the kernel of d : G^{1}(F, ℓ) ⟶ U^{1}(F, ℓ) is equal to G^{3}(F, ℓ).

Proof: The kernel of d : G^{1}(F, ℓ) ⟶ U^{1}(F, ℓ) is contained in the kernel of d_{2} : G^{2}(F, ℓ) ⟶ K_{2}^{(ℓ)}F, whence in G^{3}(F, ℓ), by (7.9). Assume now that −1 ∈ F^{×}^{ℓ}, i.e. {−1} = 0 in K_{1}^{(ℓ)}F. Since {−1} = 0 in K_{1}^{(ℓ)}F, (7.8) yields that d(π − 4 × ℍ) = 1 for any 3-fold Pfister form π over F. Using (6.5), this implies that d is trivial on G^{3}(F, ℓ). □

7.12 Corollary. The homomorphism d : G^{1}(F, ℓ) ⟶ U^{1}(F, ℓ) is injective in each of the following cases:

• ℓ is odd ;

• F^{×} = F^{×2} and ℓ > 0 ;

• char(F) = 2 and F is perfect.

Proof: In any of these cases we have −1 ∈ F^{×ℓ}, so that the previous corollary applies. Hence the kernel of d : G^{1}(F, ℓ) ⟶ U^{1}(F, ℓ) is equal to G^{3}(F, ℓ). The statement then follows, because G^{3}(F, ℓ) = 0 by (6.7). □

7.13 Remark. If d : G^{1}(F, ℓ) ⟶ U^{1}(F, ℓ) is injective, then we have immediately a positive answer to (5.4).

7.14 Proposition. The homomorphism d : G^{1}(F, ℓ) ⟶ U^{1}(F, ℓ) is surjective if and only if K_{3}^{(ℓ)}F = 0.

Proof: Given a symbol σ in K_{3}^{(ℓ)}F, it is easy to see that 1+σ lies in the image of d if and only if σ = 0. This shows that the condition in the statement is necessary.

Assume now that K_{3}^{(ℓ)}F = 0. The elements of U^{1}(F, ℓ) then are of the shape 1 + α + β with α ∈ K_{1}^{(ℓ)}F and β ∈ K_{2}^{(ℓ)}F. Let such α and β be given. There is an element ξ ∈ G^{2}(F, ℓ) such that β = d_{2}(ξ). Let d ∈ F^{×} be such that α = {d} in K_{1}^{(ℓ)}F. For ζ = ξ + ⟨d⟩ − ⟨1⟩ ∈ G^{1}(F, ℓ), we obtain

d(ζ) = d(ξ) · d(⟨d⟩ − ⟨1⟩) = (1 + β) · (1 + α) = 1 + α + β ,

since β · α ∈ K_{3}^{(ℓ)}F = 0. Therefore the condition is sufficient. □

7.15 Conjecture. If K_{3}^{(ℓ)}F = 0, then d : G^{1}(F, ℓ) ⟶ U^{1}(F, ℓ) is an isomorphism.

We finish this section by a result on interdependencies among the different Delzant classes of a form. It generalises [8, Remark 3.4].

7.16 Theorem. Let n be a positive integer. Let n_{1} > … > n_{k} be the decreasing sequence of 2-powers such that n = n_{1} + ⋯ + n_{k}. Then for any ℓ-form φ over F one has the equality

d_{n}(φ) = d_{n_{1}}(φ) ⋯ d_{n_{k}}(φ) .

Proof: We put d_{n}^{∗}(φ) = d_{n_{1}}(φ) ⋯ d_{n_{k}}(φ) and want to prove that d_{n}^{∗}(φ) = d_{n}(φ). If rk(φ) = 1 this is trivial. We proceed by induction on rk(φ).

Assume now that rk(φ) > 1 and write φ = ψ + ⟨x⟩. For any m ≥ 1, we have by the induction hypothesis

d_{m}^{∗}(ψ) = d_{m}(ψ) ,

and we know further from (7.1) and (7.2) that

d_{m}(φ) = d_{m}(ψ) + d_{m−1}(ψ) · {x} .

We compute

d_{n}(φ) = d_{n}(ψ) + d_{n−1}(ψ) · {x}
= d_{n}^{∗}(ψ) + d_{n−1}^{∗}(ψ) · {x}
= d_{n−n_{k}}^{∗}(ψ) · (d_{n_{k}}(ψ) + d_{n_{k}−1}^{∗}(ψ) · {x})
= d_{n−n_{k}}^{∗}(ψ) · (d_{n_{k}}(ψ) + d_{n_{k}−1}(ψ) · {x})
= d_{n−n_{k}}^{∗}(ψ) · d_{n_{k}}(φ)
= d_{n_{1}}(ψ) ⋯ d_{n_{k−1}}(ψ) · d_{n_{k}}(φ) .

To finish the proof, we shall show for 1 ≤ i < k that

d_{n_{i}}(ψ) · d_{n_{k}}(φ) = d_{n_{i}}(φ) · d_{n_{k}}(φ) .

Since n_{i} is a 2-power greater than n_{k}, the element ξ = d_{n_{i}−1}^{∗}(ψ) · {x} in K_{∗}^{(ℓ)}F is a multiple of d_{n_{k}−1}(ψ) · {x} as well as of d_{n_{k}}(ψ). Using that ζ^{2} = {−1}^{m} ·ζ holds for any ζ ∈ K_{m}^{(ℓ)}F and m ≥ 0, it follows that

d_{n_{k}−1}^{∗}(ψ) · {x} · ξ = {−1}^{n_{k}} · ξ = d_{n_{k}}(ψ) · ξ .

Since d_{n_{k}−1}^{∗}(ψ) = d_{n_{k}−1}(ψ), we obtain that

d_{n_{k}}(φ) · ξ = (d_{n_{k}}(ψ) + d_{n_{k}−1}(ψ) · {x}) · ξ = 0 .

Since further ξ = d_{n_{i}−1}(ψ) · {x}, we conclude that

d_{n_{i}}(φ) · d_{n_{k}}(φ) = (d_{n_{i}}(ψ) + d_{n_{i}−1}(ψ) · {x}) · d_{n_{k}}(φ) = d_{n_{i}}(ψ) · d_{n_{k}}(φ) ,

which is what we claimed above. □

8 Vanishing of higher K-groups

To illustrate how virtual forms and Delzant classes can be used to do computations in Milnor K-theory, we give an alternative proof of a result due to B. Kahn which, under the hypothesis that −1 ∈ F^{×ℓ}, relates the ‘symbol length’ of K_{2}^{(ℓ)}F to the vanishing on higher K-groups modulo ℓ.

8.1 Proposition. Let ξ ∈ K_{2}^{(ℓ)}F and m ∈ ℕ. The following are equivalent:

(i) ξ is a sum of m symbols in K_{2}^{(ℓ)}F.

(ii) There exists an ℓ-form φ of rank 2m+2 over F such that ξ = d_{2}(φ−φ^{∘}).

(iii) There exists an ℓ-form φ of rank 2m+1 and of determinant {(−1)^{m+1}} over F such that ξ = d_{2}(φ − φ^{∘}).

Proof: The implication (iii ⇒ ii) is obvious.

(ii ⇒ i) Assume that ξ = d_{2}(φ−φ^{∘}), where φ is an ℓ-form of rank 2m+2 over F. By (4.13), there exist ℓ-forms ϑ_{1}, …, ϑ_{m} over F, each of rank 4 and of trivial determinant, such that φ−φ^{∘} = (ϑ_{1} −⟨⟨1, 1⟩⟩)+ ⋯ +(ϑ_{m} −⟨⟨1, 1⟩⟩). For i = 1, …, m we put ξ_{i} = d_{2}(ϑ_{i} − ⟨⟨1, 1⟩⟩) and obtain ξ = ξ_{1} + ⋯ + ξ_{m}. Now, if ϑ = ⟨x, y, z, (xyz)^{−1}⟩, then d_{2}(ϑ − ⟨⟨1, 1⟩⟩) = {−xy, −yz}. Therefore ξ_{1}, …, ξ_{m} are symbols in K_{2}^{(ℓ)}F, so ξ is a sum of m symbols.

(i ⇒ iii) Assume that ξ is a sum of m symbols in K_{2}^{(ℓ)}F. We show by induction on m that there exists an ℓ-form φ of rank 2m + 1 and of determinant d_{1}(φ) = {(−1)^{m+1}} such that ξ = d_{2}(φ − φ^{∘}). If m = 0, this is trivial. We assume now that m > 0 and write ξ = ζ +{a, b}, where a, b ∈ F^{×} and ζ is a sum of m − 1 symbols in K_{2}^{(ℓ)}F. By induction hypothesis, there exists an ℓ-form ψ of rank 2m − 1 and of determinant {(−1)^{m}} such that ζ = d_{2}(ψ − ψ^{∘}). We write ψ = ψ′ + ⟨c⟩. Now we consider the ℓ-form φ = ψ′ + ⟨x, y, z⟩, where x, y, z ∈ F^{×} shall be fixed later. Clearly, φ has rank 2m + 1. In order to have d_{1}(φ) = {(−1)^{m+1}}, we need to have xyz = −c. We compute

φ − φ^{∘} = ψ′ + ⟨x, y, z⟩ − ψ^{∘} − ℍ
= ψ′ + ⟨x, y, z⟩ + ⟨(xyz)^{−1}, −xyz⟩ − ψ^{∘} − 2 × ℍ
= (ψ′ + ⟨c⟩ − ψ^{∘}) + (⟨x, y, z, (xyz)^{−1}⟩ − ⟨⟨1, 1⟩⟩
= (ψ − ψ^{∘}) + (⟨x, y, z, (xyz)^{−1}⟩ − ⟨⟨1, 1⟩⟩)

and obtain then

d_{2} (φ − φ^{∘}) = d_{2} (ψ − ψ^{∘})+d_{2} (⟨x, y, z, (xyz)^{−1}⟩ − ⟨⟨1, 1⟩⟩) = ζ+{−xy, −yz} .

Hence, d_{2}(φ − φ^{∘}) = ξ holds as soon as we have {−xy, −yz} = {a, b}. Together with the condition that xyz = −c, this can be achieved by putting x = −b^{−1}c, y = abc^{−1} and z = a^{−1}c. □

8.2 Theorem (Kahn). Let m ∈ ℕ and assume that −1 ∈ F^{×}^{ℓ}. If every element of K_{2}^{(ℓ)}F is equal to a sum of m symbols, then K_{n}^{(ℓ)}F = 0 for any n ≥ 2m + 2.

Proof: It obviously suffices to prove the claim for n = 2m+2. By hypothesis, we have {−1} = 0 in K_{1}^{(ℓ)}F. As a consequence, for any ℓ-form φ of trivial determinant over F we have the equality d(φ−φ^{∘}) = d(φ). We shall use this observation below.

Suppose on the contrary that K_{n}^{(ℓ)}F ≠ 0. Hence there exist a_{1}, …, a_{n} ∈ F^{×} such that {a_{1}, …, a_{n}} ≠ 0 in K_{n}^{(ℓ)}F. We set a = (a_{1} ⋯ a_{n})^{−1} and φ = ⟨a_{1}, …, a_{n}, a⟩. Using that n is even, an easy computation yields that d_{n}(φ) = {a_{1}, …, a_{n}}.

Suppose now, that the element d_{2}(φ) ∈ K_{2}^{(ℓ)}F can be written as a sum of m symbols. By (8.1), there exists an ℓ-form ψ over F of rank 2m + 1 and of trivial determinant such that d_{2}(φ) = d_{2}(ψ). It follows that φ − φ^{∘} and ψ − ψ^{∘} represent the same class in G^{2}(F, ℓ)/G^{3}(F, ℓ). With (7.11) we conclude that d(φ) = d(ψ). In particular, we have

d_{n}(ψ) = d_{n}(φ) = {a_{1}, …, a_{n}} ≠ 0 .

However, since n = 2m + 1 = 2 this is in contradiction to (7.1). □

8.3 Remark. The original statement in [4, Section 3] is in slightly different terms. The proof given there uses divided power operations instead of virtual forms and Delzant classes.

Under the assumption that −1 ∈ F^{×}^{ℓ}, it follows from [4, Appendix A] that the commutative ring K_{∗∗}^{(ℓ)}F = ⨁_{i=0}^{∞} K_{2i}^{(ℓ)}F is endowed with a system of divided powers. That is, there is a unique collection of maps

[i] : K_{2m}^{(ℓ)}F ⟶ K_{2im}^{(ℓ)}F , x ⟼ x^{[i]},

for i, m ≥ 0, with the following properties:

(1) s^{[0]} = 1 and s^{[1]} = s,

(2) (st)^{[i]} = s^{i}t^{[i]},

(3) s^{[i]}t^{[j]} = \binom{i+j}{j}(st)^{[i+j]},

(4) (s + t)^{[k]} = ∑_{i+j=k} s^{[i]}t^{[j]},

(5) (s^{[i]})^{[j]} = \frac{(ij)!}{i!j!^{i}}s^{[ij]},

(6) s^{[i]} = 0 if s is a symbol and i ≥ 2.

It is not just by chance that both strategies apply to prove Kahn’s theorem. In fact, using the above rules for divided powers, one can easily prove the following link between divided powers and the Delzant classes of virtual forms: assuming that −1 ∈ F^{×}^{ℓ}, one has for any ℓ-form φ over F and for any i ≥ 1 the identities

d_{2i}(φ) = (d_{2}(φ))^{[i]} and d_{2i+1}(φ) = d_{1}(φ) · d_{2i}(φ).

9 Isotropy and representation

We generalise the concept of isotropy from quadratic forms to virtual forms.

Let φ be an ℓ-form over F. We say that φ is isotropic, if φ = ψ + ℍ for some ℓ-form ψ over F, otherwise we say that φ is anisotropic. Furthermore, we say that φ is hyperbolic, if φ = m × ℍ for some m ≥ 1.

Let F′/F be a field extension. Given an ℓ-form φ = ⟨a_{1}, …, a_{m}⟩ over F we denote by φ_{F′} the corresponding ℓ-form ⟨a_{1}, …, a_{m}⟩ over F′. Obviously passing from φ to φ_{E} defines a homomorphism G(F, ℓ) ⟶ G(F′, ℓ).

One may ask whether some well-known theorems about the isotropy behavior of quadratic forms under field extensions can be generalised in some way to virtual forms.

9.1 Question. Let F′/F be a finite field extension of degree prime to ℓ. Does any anisotropic ℓ-form over F remain anisotropic after extension to F′?

9.2 Question. Let ℓ be a prime and assume that F contains an ℓth root of unity. Let F′ = F ( ^{ℓ} \sqrt{d}) for some d ∈ F \ F^{×ℓ} and φ ∈ M(F, ℓ). Is it true that φ_{F′} is isotropic if and only φ contains a subform β with rk(β) = 2 and d_{1}(β) = {(−d)^{i}} with 0 ≤ i < ℓ?

In the case where ℓ = 2 ≠ char(F), both questions above have a positive answer, by Springer’s Theorem (cf. [6, Chap. VII, §2]) and by [6, Chap. VII, §3].

Let a ∈ F^{×} and let φ be an ℓ-form over F. If φ can be written as ⟨a⟩ + ψ (resp. as ψ +⟨a⟩) for some ℓ-form ψ, then we say that φ left-represents (resp. right-represents) a; it is easy to see that this is equivalent to saying that the form ⟨−a^{−1}⟩ ⊥ φ (resp. φ ⊥ ⟨−a^{−1}⟩) is isotropic. Note that an isotropic form (left- and right-)represents any element of F^{×}.

9.3 Proposition. Let φ be an ℓ-form over F and a ∈ F^{×}.

(a) If φ has even rank, then φ left-represents a if and only if it rightrepresents a^{−1}.

(b) If φ has odd rank, then φ left-represents a if and only if it rightrepresents a.

(c) If d_{1}(φ) = 0 or d_{1}(φ) = {−1}, then φ left-represents a if and only if it right-represents a.

Proof: Parts (a) and (b) follow from (3.3) and (c) is clear from (4.9). □

In view of the proposition, if φ has odd rank or trivial determinant, one may say that φ represents some element a ∈ F^{×} if it left-represents a.

Acknowledgments The author wishes to express his gratitude to Beatrix Bernauer, Emmanuel Lequeu, Susanne Pumplün, Thomas Unger, and Sven Wagner for their comments and suggestions on preliminary versions of this manuscript. He further acknowledges the financial support provided by the European RTN Network ‘Algebraic K-Theory, Linear Algebraic Groups, and Related Structures’ (HPRN-CT-2000-00287) and by the Swiss National Science Foundation (Grant No. 200020-100229/1).

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Karim Johannes Becher, Fachbereich Mathematik und Statistik, D203, Universität Konstanz, 78457 Konstaz, Germany. Email: becher@maths.ucd.ie

April 18, 2006