to appear in Izvestiya Math. 2013

Hasse principle for G–trace forms

E. Bayer–Fluckiger, R. Parimala, J–P. Serre

À Igor Rostislavovich Shafarevich, à l’occasion de son 90-ième anniversaire

Introduction

Let k be a field, let G be a finite group and let L be a G-Galois algebra over k , for instance a finite Galois extension of k with Galois group G. The trace form of L is the symmetric bilinear form

q_{L} : L × L → k, where q_{L}(x, y) = Tr_{L/k}(xy). Note that (L, q_{L}) is a G–quadratic space, i.e. we have q_{L}(gx, gy) = q_{L}(x, y)

for all g ∈ G and all x, y ∈ L.

An interesting special case is when L has a self-dual normal basis, i.e. when there exists x ∈ L such that (gx)_{g∈G} is a k-basis of L and q_{L}(gx, hx) = δ_{g,h} for every g, h ∈ G, where δ_{g,h} is the Kronecker symbol.

The aim of the present paper is to prove (cf. th.1.3.1): Theorem - Suppose that k is a global field of characteristic ≠ 2. Let L and L′ be two G–Galois algebras. If the G–quadratic spaces (L, q_{L}) and (L′, q_{L′}) become isomorphic over all the completions of k, then they are isomorphic over k. Corollary - If L has a self–dual normal basis over all the completions of k, then it has a self-dual normal basis over k. [In short: the Hasse principle applies to self-dual normal bases in characteristic ≠ 2.]

These results complete those of [BFP 11] where it was assumed that G has the “odd determinant property”. The present proof follows a similar pattern. We give it in §1; the only new ingredient is a restriction-induction property which is proved in §4 by using Burnside rings, following a method introduced by A. Dress in the 1970s. There are also two basic results which have to be used:

• Witt cancellation theorem for G-quadratic spaces;

• the “odd does not count” principle, which allows division by any odd integer. Their proofs are summarized (resp. completed) in §2 (resp. in §3), in the

more general setting of hermitian spaces over an algebra with involution in characteristic ≠ 2.

Acknowledgements. We thank H.W. Lenstra, Jr, and J-P. Tignol for their help with §3.4 and §3.5.

§1. G–trace forms and Hasse principle

The aim of the present § is to prove the Hasse principle result stated in the introduction, assuming the technical results of §§2,3,4.

Before doing so, we introduce some notation and basic facts concerning G– quadratic forms and G–trace forms.

In what follows, the characteristic of the ground field k is assumed to be ≠ 2.

1.1. G–quadratic spaces

Let G be a finite group. A G–quadratic space is a pair (V, q), where V is a k[G]–module that is a finite dimensional k–vector space, and q : V × V → k is a non–degenerate symmetric bilinear form such that

q(gx, gy) = q(x, y)

for all x, y ∈ V and all g ∈ G.

Two G–quadratic spaces (V, q) and (V′, q′) are isomorphic if there exists an isomorphism of k[G]–modules f : V → V′ such that q′(f(x), f(y)) = q(x, y) for all x, y ∈ V. If this is the case, we write (V, q) ≃_{G} (V′, q′).

Theorem 1.1.1 ([BFL 90, th. 4.1]) - If two G–quadratic spaces become isomorphic over an odd degree extension, then they are isomorphic.

Let S be a subgroup of G. We have two operations, induction and restriction (see for instance [BFS 94, §1.2]) :

• If (V, q) is an S–quadratic form, then Ind_{S}^{G}(V, q) is a G–quadratic form;

• If (V, q) is a G–quadratic form, then Res_{S}^{G}(V, q) is an S–quadratic form.

The following result is one of the basic tools in the proof of the Hasse principle :

Theorem 1.1.2 - Let S be a 2–Sylow subgroup of G, and let (V_{1}, q_{1}) and (V_{2}, q_{2}) be two S–quadratic spaces over k. Suppose that

Res_{S}^{G} Ind_{S}^{G}(V_{1}, q_{1}) ≃_{S} Res_{S}^{G} Ind_{S}^{G}(V_{2}, q_{2}).

Then

Ind_{S}^{G}(V_{1}, q_{1}) ≃_{G} Ind_{S}^{G}(V_{2}, q_{2}).

This was proved in [BFS 94, th.5.3.1] when S is elementary abelian and in [BFP 11, theorem 2.2] when the fusion of S in G is controlled by the normalizer of S. The proof in the general case will be given in §4.3.

Note that the Hasse principle does not hold in general for G–quadratic spaces (see e.g. [M 86, §2]). However, it does hold when G is a 2–group :

Proposition 1.1.3 - Suppose that k is a global field, and let S be a 2–group. If two S–quadratic spaces become isomorphic over all the completions of k, then they are isomorphic over k.

Proof. By a theorem of M. Kneser (see e.g. [R 11, th.3.3.3]), this is true for every finite group S, of order prime to the characteristic of k, whose group algebra k[S] has the following property:

(P) - No simple factor of k[S], stable under the canonical involution of k[S], is a matrix algebra over a quaternion skewfield with an orthogonal involution.

By [BFP 11, prop.3.7], property (P) holds when S is a 2-group and the characteristic of k is not 2.

1.2. Trace forms

Let L be a G–Galois algebra over k. Recall (cf. e.g. [BFS 94, §1.3] or [A8, §16.7]) that this means that L is an étale k-algebra on which G acts by k-automorphisms in such a way that L is a free k[G]-module of rank 1. Let

q_{L} : L × L → k, q_{L}(x, y) = Tr_{L/k}(xy)

be the trace form of L. Then (L, q_{L}) is a G–quadratic space.

Lemma 1.2.1 - Let S be a 2–Sylow subgroup of G. If L is a G–Galois algebra, there exists an odd degree field extension k′/k and an S–Galois algebra M over k′ such that the G–form (L, q_{L})⊗_{k} k′ is isomorphic to the G–form Ind_{S}^{G}(M, q_{M}). This is proved in [BFS 94, prop.2.1.1].

1.3. Proof of the Hasse principle

We now prove the result stated in the introduction:

Theorem 1.3.1 - Suppose that k is a global field. Let L_{1} and L_{2} be two G–Galois algebras. If the G–quadratic spaces (L_{1}, q_{L_{1}}) and (L_{2}, q_{L_{2}}) become isomorphic over all the completions of k, then they are isomorphic over k.

Proof. By lemma 1.2.1, there exists an odd degree field extension k′/k and two S–Galois algebras M_{1} and M_{2} over k′ such that

(L_{1}, q_{L_{1}}) ⊗_{k} k′ ≃_{G} Ind_{S}^{G}(M_{1}, q_{M_{1}}) and (L_{2}, q_{L_{2}}) ⊗_{k} k′ ≃_{G} Ind_{S}^{G}(M_{2}, q_{M_{2}}).

Suppose that the G–quadratic spaces (L_{1}, q_{L_{1}}) and (L_{2}, q_{L_{2}}) are isomorphic over all the completions of k. This implies that the G–quadratic spaces

(L_{1}, q_{L_{1}}) ⊗_{k} k′ and (L_{2}, q_{L_{2}}) ⊗_{k} k′

are isomorphic over all the completions of k′. Therefore the S–quadratic spaces

Res_{S}^{G}(L_{1}, q_{L_{1}}) ⊗_{k} k′ ≃_{S} Res_{S}^{G} Ind_{S}^{G}(M_{1}, q_{M_{1}})

and

Res_{S}^{G}(L_{2}, q_{L_{2}}) ⊗_{k} k′ ≃_{S} Res_{S}^{G} Ind_{S}^{G}(M_{2}, q_{M_{2}})

are isomorphic over all the completions of k′. Since the Hasse principle holds for S–quadratic spaces (cf. prop.1.1.3), this implies that

Res_{S}^{G} Ind_{S}^{G}(M_{1}, q_{M_{1}}) ≃_{S} Res_{S}^{G} Ind_{S}^{G}(M_{2}, q_{M_{2}}).

By th.1.1.2 we have

Ind_{S}^{G}(M_{1}, q_{M_{1}}) ≃_{G} Ind_{S}^{G}(M_{2}, q_{M_{2}}).

Recall that we have

(L_{1}, q_{L_{1}}) ⊗_{k} k′ ≃_{G} Ind_{S}^{G}(M_{1}, q_{M_{1}}) and (L_{2}, q_{L_{2}}) ⊗_{k} k′ ≃_{G} Ind_{S}^{G}(M_{2}, q_{M_{2}}),

hence

(L_{1}, q_{L_{1}}) ⊗_{k} k′ ≃_{G} (L_{2}, q_{L_{2}}) ⊗_{k} k′.

By th.1.1.1, we get

(L_{1}, q_{L_{1}}) ≃_{G} (L_{2}, q_{L_{2}}),

as claimed.

§2. Witt’s cancellation theorem for ϵ–hermitian spaces

Witt’s cancellation theorem applies not only to quadratic forms, but also to hermitian spaces over finite dimensional algebras with involution, provided the characteristic of the ground field is ≠ 2. This is due to H-G. Quebbemann, R. Scharlau, W. Scharlau and M. Schulte, cf. [QSSS 76, 3.4 (i)]; see also [QSS 79, 3.4.(iii)] and [K 91 chap.II, th.6.6.1]. In what follows we fix the notation and we summarize the different steps of the proof.

2.1. ϵ–hermitian spaces

From now on, unless otherwise stated, all modules are left modules.

Let R be a ring endowed with an involution r ↦ \overline{r}. For any R–module M, we denote by M^{∗} its dual Hom_{R}(M, R). Then M^{∗} has an R–module structure given by (rf)(x) = f(x)\overline{r} for all r ∈ R, x ∈ M and f ∈ M^{∗}. If M and N are two R–modules and if f : M → N is a homomorphism of R–modules, then f induces a homomorphism f^{∗} : N^{∗} → M^{∗} defined by f^{∗}(g) = gf for all g ∈ N^{∗}.

Let ϵ = ±1 and let M be an R-module. An ϵ–hermitian form on M is a biadditive map h : M × M → R which satisfies the following condition: h(rx, sy) = rh(x, y)\overline{s} and \overline{h(x, y)} = ϵh(y, x) for all x, y ∈ M and all r, s ∈ R.

In that case, the map h : M → M^{∗} given by y ↦ h( , y) is an R–homomorphism. If it is an isomorphism, we say that h is nonsingular, and we call the pair (M, h) an ϵ–hermitian space.

If (M, h) and (M′, h′) are two ϵ–hermitian spaces, their orthogonal sum is also an ϵ–hermitian space; it is denoted by (M, h) ⊕ (M′, h′). Remark on notation

If (M, h) is an ϵ–hermitian space, we are using the letter h for two different, but closely related, objects: a biadditive map M × M → R, and an R–linear homomorphism M → M^{∗}. There is usually no risk of confusion, but, when there is one, we shall write h_{b} for the biadditive map M × M → R and keep h for the map M → M^{∗}. Example : the group algebra k[G]

Let k be a commutative ring, and let G be a finite group. The group algebra R = k[G] has a natural k–linear involution, characterized by the formula \overline{g} = g^{−1} for every g ∈ G. We have the following dictionary:

a) R–module M ⟺ k-module M with a k–linear action of G;

b) R–dual M^{∗} ⟺ k–dual of M, with the contragredient (i.e. dual) action of G.

c) ϵ–hermitian space (M, h) ⟺ ϵ–symmetric bilinear form on M, which is G–invariant and defines an isomorphism of M onto its k–dual. [Let us be more specific about these identifications. Denote by α : k[G] → k the linear form “coefficient of 1”, i.e. α(g) = 1 if g = 1 and α(g) = 0 if g ≠ 1. The identification of b) transforms y ∈ M^{∗} = Hom_{R}(M, R) into the k–linear form α ∘ y. Similarly, the identification of c) transforms h_{b} into the k–bilinear form (x, y) ↦ α(h_{b}(x, y)).]

Note that, when k is a field, condition c) implies that M is finite dimensional over k (if not it would not be isomorphic to its dual), so that, when ϵ = 1, an ϵ–hermitian space over k[G] is the same as what we called a G–quadratic space in §1.1.

2.2. Basic hypotheses for §2 and §3

In the rest of §2 and in §3, we assume that R is an algebra over a field k and that its involution r ↦ \overline{r} is k–linear. We also assume:

(2.2.1) The characteristic of k is ≠ 2.

(2.2.2) dim_{k} R < ∞. [The only case we actually need for theorem 1.3.1 is R = k[G], with G finite.]

Note that (2.2.2) implies that R is a left and right artinian ring, cf. [A8, §1]. If (M, h) is an ϵ–hermitian space over R, we shall always assume:

(2.2.3) dim_{k}M < ∞.

Because of (2.2.2), such an M is an R–module of finite length, so that one may apply the Krull-Remak-Schmidt theory to it; in particular the cancellation theorem holds, cf. [A 8, §2.4, cor.3 to th.1].

2.3. Biduality and reflexivity

If M is an R-module, the dual (M^{∗})^{∗} of its dual is called its bidual and is denoted by M^{∗∗}. Every element x ∈ M defines an R-linear form ℓ_{x} on M^{∗} by the formula ℓ_{x}(f) = \overline{f(x)}, for f ∈ M^{∗}. The map x ↦ ℓ_{x} is an Rlinear homomorphism M → M^{∗∗}, which we shall denote by c_{M}. The same construction applies to M^{∗}, so that we have c_{M^{∗}} : M^{∗} → M^{∗∗∗}. On the other hand, the adjoint (c_{M})^{∗} of c_{M} is a homomorphism of M^{∗∗∗} into M^{∗}. These two maps are related by : Proposition 2.3.1 - The composition

(c_{M})^{∗} ∘ c_{M^{∗}} : M^{∗} → M^{∗∗∗} → M^{∗}

is the identity.

The proof is a direct computation. To explain it, it is convenient to denote the duality pairing between M and M^{∗} by a scalar product, i.e. to write <x, y > instead of y(x), for x ∈ M and y ∈ M^{∗}. With a similar convention for the other modules, one finds:

<x, (c_{M})^{∗} ∘ c_{M^{∗}}(y)> = <c_{M}(x), c_{M^{∗}}(y)> = <c_{M}(x), y > = <x, y >, and since this is valid for every x, we have (c_{M})^{∗} ∘c_{M^{∗}}(y) = y for every y ∈ M^{∗}, as claimed.

A module M is said to be reflexive if c_{M} : M → M^{∗∗} is an isomorphism; it is said to be selfdual if M and M^{∗} are isomorphic. Proposition 2.3.2 - (1) If M ≃ M_{1} ⊕ M_{2}, then M is reflexive if and only if both M_{1} and M_{2} are reflexive.

(2) If a module N is isomorphic to the dual of some module, then c_{N} is injective and its image is a direct factor in N^{∗∗}.

(3) If M is selfdual, M is reflexive.

(4) If M is reflexive, so is M^{∗}.

Proof. Assertion (1) is immediate. Assertion (2) follows from prop.2.3.1 applied to a module M such that N ≃ M^{∗}. In case (3), M^{∗} is isomorphic to M; hence M^{∗∗} ≃ M^{∗} ≃ M, and the modules M and M^{∗∗} have the same length; since c_{M} : M → M^{∗∗} injective (by (2)), it is an isomorphism. In case (4), c_{M^{∗}} is injective because of (2), and since M^{∗} ≃ M^{∗∗}, the length argument above shows that c_{M^{∗}} is bijective.

Corollary 2.3.3 - If (M, h) is an ϵ–hermitian space, then M is reflexive, and so are all its direct factors.

Since M is selfdual, this follows from parts (3) and (1) of prop.2.3.2.

2.4. Hyperbolic hermitian spaces

Let ϵ = ±1, and let N be a reflexive module. Define an ϵ–hermitian form h_{N} on N ⊕ N^{∗} by the biadditive map

((x, y), (x′, y′)) ↦ y′(x) + ϵy(x′) for x, x′ ∈ N and y, y′ ∈ N^{∗}. Since N is reflexive, h_{N} is nonsingular. The hermitian space (N ⊕ N^{∗}, h_{N}) is denoted by H_{ϵ}(N). Note that H_{ϵ}(N) ≃ H_{ϵ}(N^{∗}).

An ϵ–hermitian space (M, h) is called hyperbolic if there exists a reflexive module N such that (M, h) ≃ H_{ϵ}(N).

This is equivalent to asking that the module M splits as a direct sum M = N ⊕ N′ where N and N′ are totally isotropic, i.e. h_{b}(x, y) = 0 when x, y ∈ N and when x, y ∈ N′. Example. If (M, h) is an ϵ–hermitian space, then (M, h)⊕(M, −h) is hyperbolic: apply the criterion to N = diagonal of M ⊕M and N′ = antidiagonal of M ⊕M.

Proposition 2.4.1 - Let (M, h) and (M′, h′) be two ϵ–hermitian hyperbolic modules. If M ≃ M′, then (M, h) ≃ (M′, h′). Lemma 2.4.2 - Let N and N′ be two reflexive modules such that N ⊕ N^{∗} ≃ N′ ⊕N^{′∗}. Then there exists two reflexive modules P and Q such that N ≃ P ⊕Q and N′ ≃ P ⊕ Q^{∗}. Proof of the lemma. We use induction on the length ℓ of N. The case ℓ = 0 is trivial. If ℓ > 0, we may decompose N as S⊕N_{0}, where S is indecomposable and the length of N_{0} is < ℓ. By the Krull-Remak-Schmidt theorem, S is isomorphic to a direct factor of either N′ or N^{′∗}. In the first case, we have N′ ≃ S ⊕ N′_{0} for some reflexive module N′_{0}, hence

S ⊕ N_{0} ⊕ S^{∗} ⊕ N_{0}^{∗} ≃ S ⊕ N′_{0} ⊕ S^{∗} ⊕ (N′_{0})^{∗}. By the cancellation theorem for modules, this implies N_{0} ⊕ N_{0}^{∗} ≃ N′_{0} ⊕ (N′_{0})^{∗}, and we apply the induction assumption to the pair (N_{0}, N′_{0}). The second case follows from the first one, applied to the pair (N, N^{′∗}). Proof of proposition 2.4.1. Suppose that (M, h) ≃ H_{ϵ}(N) and (M′, h′) ≃ H_{ϵ}(N′). We have N ⊕ N^{∗} ≃ N′ ⊕ N^{′∗}. By the lemma above, we may write N as P ⊕ Q and N′ as P ⊕ Q^{∗}. We then have:

(M, h) ≃ H_{ϵ}(N) ≃ H_{ϵ}(P) ⊕ H_{ϵ}(Q) ≃ H_{ϵ}(P) ⊕ H_{ϵ}(Q^{∗}) ≃ H_{ϵ}(N′) ≃ (M′, h′), as claimed.

2.5. The cancellation theorem

Let Ref_{R} be the category of all reflexive finitely generated R–modules. This is an additive category, with a duality functor M ↦ M^{∗}, as in [K 91, chap.II, §6.2]. Proposition 2.5.1 - The category Ref_{R} has the properties C.1, C.2 and C.3 of [K 91, chap.II, §5.2 and §6.2]. Proof. Property C.1 asks that every idempotent splits; this follows from prop.2.3.2 (1).

Property C.2 has two parts. First, every object M should split as a finite direct sum of indecomposable objects; this is true because R is an artinian ring. Second, if M is indecomposable, the ring A = End(M) should be a (non necessarily commutative) local ring, i.e. A should be ≠ 0 and, for all a ∈ A, either a or 1 − a should be invertible; this is true, cf. [A 8, §2.5, th.2].

Property C.3 asks that, for every object M, the ring A = End(M) should be complete for the topology defined by the powers of its radical; indeed, A is artinian (because it is a finitely generated k–module), hence its radical is nilpotent ([A 8, §10.1, prop.1]), and the corresponding topology is the discrete topology.

We may thus apply to Ref_{R} the results proved by [QSSS 76], [QSS 79] and [K 91 Chap.II, §6]. The main one, for our purposes, is the cancellation theorem:

Theorem 2.5.2 - Let (M, h), (M′, h′) and (N, g) be ϵ–hermitian spaces such that :

(M, h) ⊕ (N, g) ≃ (M′, h′) ⊕ (N, g).

Then (M, h) ≃ (M′, h′). [Of course it is assumed that R and the R–modules M, M′, N satisfy the conditions in §2.2, so that M, M′ and N belong to the category Ref_{R}.] Proof. This is a special case of theorem 6.6.1 of [K 91, chap.II]. The proof given there has two parts:

a) when M, M′ and N are isotypic of the same self-dual type, by reduction to the standard Witt cancellation theorem for hermitian spaces over division algebras, cf. e.g. [A9, §4.3, th.1] or [K 91, chap.I, th.6.5.2].

b) in the general case, by using the isotypic decompositions of hermitian spaces, cf. [K 91, chap.II, th.6.3.1].

2.6. Witt groups and Grothendieck-Witt groups

The Grothendieck-Witt group WGr^{ϵ}(R) is defined as the Grothendieck group of the category of the ϵ–hermitian spaces over R (having property (2.2.3), of course), with respect to the orthogonal direct sum. If (M, h) is an ϵ–hermitian space, its class in WGr^{ϵ}(R) will be denoted by [M, h]_{gr}. Because of theorem 2.5.2, we have:

(2.6.1) (M, h) ≃ (M′, h′) ⟺ [M, h]_{gr} = [M′, h′]_{gr} .

Let Hyp_{R}^{ϵ} be the subgroup of WGr^{ϵ}(R) generated by the classes of the hyperbolic spaces (cf. §2.4). The quotient

W^{ϵ}(R) = WGr^{ϵ}(R)/Hyp_{R}^{ϵ} is called the Witt group of (R, ϵ). If (M, h) is an ϵ–hermitian space, its class in W^{ϵ}(R) will be denoted by [M, h].

Proposition 2.6.1 - Let (M, h) and (M′, h′) be two ϵ–hermitian spaces. If M ≃ M′ and [M, h] = [M′, h′], then (M, h) ≃ (M′, h′). Proof. Since [M, h] = [M′, h′], there exist two hyperbolic ϵ–hermitian spaces (N, q) and (N′, q′) such that

(M, h) ⊕ (N, q) ≃ (M′, h′) ⊕ (N′, q′). This implies M ⊕ N ≃ M′ ⊕ N′, and since M ≃ M′, we have N ≃ N′; by prop.2.4.1, we have (N, q) ≃ (N′, q′), hence (M, h) ≃ (M′, h′) by theorem 2.5.2.

Proposition 2.6.2 - (i) The subgroup Hyp_{R}^{ϵ} of WGr^{ϵ}(R) is torsion-free.

(ii) Let π : WGr^{ϵ}(R) → Hyp_{R}^{ϵ} be the homomorphism which maps the class of (M, h) to the class of the hyperbolic space H_{ϵ}(M). Then π(x) = 2x for every x ∈ Hyp_{R}^{ϵ}. Proof of (i). Let x ∈ Hyp_{R}^{ϵ} be such that nx = 0, with n > 0. There exist reflexive modules N and N′ such that x = [H(N)]_{gr} − [H(N′)]_{gr}; the relation nx = 0 means that H(n•N) ≃ H(n•N′), where n•N (resp. n•N′) denotes the direct sum of n copies of N (resp. of N′). This implies n • (N ⊕ N^{∗}) ≃ n • (N′ ⊕ N^{′∗}); by the Krull-Remak-Schmidt theory for modules, we have N ⊕ N^{∗} ≃ N′ ⊕ N^{′∗}, hence H(N) ≃ H(N′) by proposition 2.4.1; hence x = 0. Proof of (ii). It is enough to prove the formula when x is the class of a hyperbolic space H_{ϵ}(N); in that case π(x) is the class of H_{ϵ}(N ⊕ N^{∗}) = H_{ϵ}(N) ⊕ H_{ϵ}(N), which is 2x.

Proposition 2.6.3 - Let n be an integer > 0. The following properties are equivalent :

(a) WGr^{ϵ}(R) is n–torsion free.

(b) If (M, h) and (M′, h′) are two ϵ–hermitian spaces such that n • (M, h) ≃ n • (M′, h′), then (M, h) ≃ (M′, h′).

If n is odd, these properties are also equivalent to :

(c) W^{ϵ}(R) is n–torsion free. [Recall that an abelian group A is called n–torsion free if a ∈ A and na = 0 imply a = 0.] Proof of (a) ⟺ (b). This follows from (2.6.1). Proof of (a) ⟹ (c). We have to show that, if x ∈ WGr^{ϵ}(R) and nx ∈ Hyp_{R}^{ϵ}, then x belongs to Hyp_{R}^{ϵ}. By part (ii) of proposition 2.6.2 we have π(nx) = 2nx, hence n(2x − π(x)) = 0. Since WGr^{ϵ}(R) is n–torsion free, this shows that 2x = π(x). Hence 2x belongs to Hyp_{R}^{ϵ}; since nx has the same property, and n is odd, we have x ∈ Hyp_{R}^{ϵ}. Proof of (c) ⟹ (a). Let x ∈ WGr^{ϵ}(R) be such that nx = 0. Since W^{ϵ}(R) is n–torsion free, we have x ∈ Hyp_{R}^{ϵ}, and, by part (i) of proposition 2.6.2, we have x = 0.

In the next §, we shall see that properties (a), (b) and (c) hold for every odd n and for every (R, ϵ).

§3. Division properties for ϵ–hermitian spaces

We keep the notation of §2. In particular, k is a field of characteristic ≠ 2, R is a finite dimensional k–algebra with a k–linear involution x ↦ \overline{x}, and ϵ = ±1.

3.1. Statement of the division theorem

Let (M, h) be an ϵ–hermitian space over R, and let n be an integer > 0. We say that (M, h) has property Div_{n} if every (M′, h′) with n•(M′, h′) ≃ n•(M, h) is isomorphic to (M, h), i.e. if one is allowed to “ divide by n ”. [Recall that n • (M, h) is the direct sum of n copies of (M, h).] Note that the hypothesis n • (M′, h′) ≃ n • (M, h) implies M ≃ M′ by the Krull-Remak-Schmidt theorem for modules; hence we may assume that M′ = M, so that the problem consists in comparing two different ϵ-hermitian forms on the same R-module.

We say that R has property Div_{n,ϵ} if every ϵ–hermitian space over R has property Div_{n}. The goal of §3 is to prove: Theorem 3.1.1 - Let n be a positive odd integer. Then :

(a) The group WGr^{ϵ}(R) is n–torsion free.

(b) The group W^{ϵ}(R) is n–torsion free.

(c) Property Div_{n,ϵ} holds.

By proposition 2.6.3 the three assertions (a), (b) and (c) are equivalent. Their proof will consist in a series of reductions.

Remark. Division by 2 is rarely possible. For instance, if k = Q, the quadratic forms x^{2} and 2x^{2} are not isomorphic, but they become so after duplication:

x^{2} + y^{2} ≃ (x + y)^{2} + (x − y)^{2} = 2x^{2} + 2y^{2}. However, we shall see later (theorem 3.5.3) that, if k is real closed, then WGr^{ϵ}(R) is torsion-free, so that property Div_{n,ϵ} holds for every n > 0, including n = 2.

3.2. The W(k)–module structure of W^{ϵ}(R)

Let (M, h) be an ϵ–hermitian space over R, and let V be a finite-dimensional k–vector space equipped with a nonsingular symmetric bilinear form q (note that (V, q) may be viewed as a 1–hermitian space over k). The tensor product V ⊗_{k} M has a natural structure of ϵ–hermitian space over R, which we denote by (V, q) ⊗_{k} (M, h): this is easily checked, using the fact that the R-dual of V ⊗_{k} M is canonically isomorphic to V′ ⊗_{k} M^{∗}, where V′ is the k–dual of V.

Let WGr(k) denote the Witt-Grothendieck ring of the field k. Using the construction above, we see that WGr^{ϵ}(R) has a natural structure of WGr(k)- module. After dividing by hyperbolic spaces, this shows that W^{ϵ}(R) has a natural structure of W(k)-module. We shall use this structure to prove:

Proposition 3.2.1 - Suppose that k is not formally real. Then :

(a) There exists r ∈ N such that 2^{r}.W^{ϵ}(R) = 0.

(b) Property Div_{n,ϵ} holds for every odd n and every ϵ = ±1. [Recall that k is formally real if −1 is not a sum of squares in k; this is equivalent to the existence of a total order on k which is compatible with the field structure, cf. [A6, §2.2]; such a field has characteristic 0.] Proof. The hypothesis on k implies that there exists r ∈ N such that 2^{r} = 0 in W(k), cf. e.g. [S 85, chap.II, th.6.4 (ii)]. Since W^{ϵ}(R) is a W(k)–module, we have 2^{r}x = 0 for every x ∈ W^{ϵ}(R). This proves (a). As for (b), it follows from (a) and from proposition 2.6.3, since W^{ϵ}(R) is n-torsion free.

Proposition 3.2.1 allows a first reduction: it is enough to prove theorem 3.1.1 when k is formally real. Before we do the second reduction, we need to recall some elementary (but not completely obvious) properties of hermitian spaces with respect to field extensions.

3.3. Field extensions : going up

Let K be a field extension of k and let R_{K} = K ⊗_{k} R. If M is an R-module, let M_{K} denote the R_{K}-module K ⊗_{k} M. Let M^{∗} = Hom_{R}(M, R) be the dual of M and let (M_{K})^{∗} = Hom_{R_{K}} (M_{K}, R_{K}) be the dual of M_{K}. Every element of M^{∗} extends lineary to an element of (M_{K})^{∗} and we obtain in this way a natural map

ι_{M} : K ⊗ M^{∗} → (M_{K})^{∗}.

Proposition 3.3.1 - If M is a finitely generated R-module, ι_{M} is an isomorphism. Proof. Let a and b denote the functors M ↦ K ⊗ M^{∗} and M ↦ (M_{K})^{∗}. When M = R, we have a(M) = b(M) = R_{K} and ι_{M} is the identity. By additivity, this shows that the proposition is true when M is R-free. In the general case, we choose an exact sequence L′ → L → M → 0, where L′ and L are finitely generated R-free modules; this is possible since M is finitely generated and the ring R is noetherian. Since both a and b are left exact functors, we have a commutative diagram where the vertical maps are 0, ι_{M}, ι_{L} and ι_{L′} :

0 → a(M) → a(L) → a(L′)
↓ ↓ ↓ ↓
0 → b(M) → b(L) → b(L′).

Since ι_{L} and ι_{L′} are isomorphisms, so is ι_{M}.

The proposition above allows us to identify K ⊗ M^{∗} with (M_{K})^{∗} and to denote both modules by M_{K}^{∗}. Let now (M, h) be an ϵ-hermitian space over R. The isomorphism h : M → M^{∗} gives, by extension of scalars, an isomorphism h_{K} : M_{K} → M_{K}^{∗}. We thus obtain an ϵ-hermitian space (M_{K}, h_{K}) over R_{K}. The functor (M, h) ↦ (M_{K}, h_{K}) transforms hyperbolic spaces into hyperbolic spaces. Hence we get natural maps

WGr^{ϵ}(R) → WGr^{ϵ}(R_{K}) and W^{ϵ}(R) → W^{ϵ}(R_{K}).

In [S 85, chap.2, §5], these maps are denoted by r_{K/k}^{∗}. We shall simplify the notation by writing them x ↦ x_{K}.

3.4. Field extensions : going down

In this section, K is a finite field extension of k. We choose a non-zero k-linear map s : K → k. The Scharlau transfer for vector spaces

Let M be a vector space over K, and let N be a finite dimensional vector space over k. Let N_{K} = K ⊗_{k} N; the map s defines s_{N} = s ⊗ 1 : N_{K} → N. If f : M → N_{K} is K-linear, the composition s_{N} ∘ f is a k-linear map M → N. We thus get a map

s_{∗} : Hom_{K}(M, N_{K}) → Hom_{k}(M, N).

Proposition 3.4.1 - The map s_{∗} : Hom_{K}(M, N_{K}) → Hom_{k}(M, N) is an isomorphism. Proof. Put a(M, N) = Hom_{K}(M, N_{K}) and b(M, N) = Hom_{k}(M, N). Both functors transform direct sums (for M) into direct products, and direct sums (for N) into direct sums. Hence it is enough to check that s_{∗} is an isomorphism when M and N are 1-dimensional, i.e. when M = K and N = k; in that case, both a(M, N) and b(M, N) are K-vector spaces of dimension 1, and the map is K-linear and non zero; hence it is an isomorphism. Remark. When K = C, k = R and N = k, the proposition amounts to the standard fact that every R-linear form on a complex vector space is the real part of a unique C-linear form. The Scharlau transfer for modules

We keep the same notation (K, s, N, M) as above, and we assume that both N and M are modules over a k-algebra A; in that case, M is an A_{K}- module, where A_{K} = K ⊗_{k} A. We have natural inclusions: Hom_{A_{K}} (M, N_{K}) ⊂ Hom_{K}(M, N_{K}) and Hom_{A_{k}}(M, N) ⊂ Hom_{k}(M, N). The map s_{∗} of proposition 3.4.1 maps the first subspace into the second one. Hence we get s_{∗,A} : Hom_{A_{K}} (M, N_{K}) → Hom_{A}(M, N).

Proposition 3.4.2 - The map s_{∗,A} is an isomorphism. Proof. The injectivity of s_{∗,A} follows from that of s_{∗}. For the surjectivity, it is enough to show that, if f : M → N_{K} is a K-linear map such that s_{N} ∘ f is A-linear, then so is f. Indeed, if a is any element of A, the two maps x → f(ax) and x → af(x) have the same image by s_{∗}, hence they coincide.

The Scharlau transfer for hermitian spaces

We apply the above to A = R. If (M, h) is an ϵ-hermitian space over R_{K}, the K-bilinear map h_{b} : M × M → R_{K}, composed with s_{R} : R_{K} → R, gives an ϵ-hermitian form s_{∗} ∘ h_{b} on M, where M is now viewed as an R-module. Proposition 3.4.3 - The hermitian form s_{∗} ∘ h is nonsingular. Proof. We have to show that the corresponding map M → Hom_{R}(M, R) is an isomorphism. This map factors in:

M → Hom_{R_{K}} (M, R_{K}) → Hom_{R}(M, R). The left side map is an isomorphism because h is nonsingular; the right side

map is an isomorphism because of proposition 3.4.2. Hence the composite is an isomorphism.

We thus obtain an ϵ-hermitian space over R, which is called the Scharlau transfer of (M, h), relative to s. We denote it by (M, s_{∗}h). We use a similar notation for the corresponding homomorphisms

s_{∗} : WGr^{ϵ}(R_{K}) → WGr^{ϵ}(R) and s_{∗} : W^{ϵ}(R_{K}) → W^{ϵ}(R).

A basic formula

The definitions above apply to R = k and they give a map s_{∗} : W(K) → W(k) between the corresponding Witt groups. The element s_{∗}(1) of W(k) is the Witt class of the k-quadratic form z ↦ s(z^{2}) on the k-vector space K. Proposition 3.4.4 - Let x ∈ W^{ϵ}(R) and let x_{K} be its image in W^{ϵ}(R_{K}), cf. §3.3. We have :

s_{∗}(x_{K}) = s_{∗}(1).x in W^{ϵ}(R). [The product of s_{∗}(1) and x makes sense, because of the W(k)-module structure of W^{ϵ}(R), cf. §3.2.] Proof. It is enough to check the formula when x is the class of an ϵ-hermitian space (M, h). In that case, the k-bilinear map M_{K} × M_{K} → R corresponding to s_{∗}(M_{K}, h_{K}) is given by:

(z ⊗ m) × (z′ ⊗ m′) ↦ s(zz′)h(m, m′) for z, z′ ∈ K and m, m′ ∈ M.

This is the same formula as for (K, q_{s}) ⊗ (M, h), where q_{s} is the quadratic form z ↦ s(z^{2}) on K.

An application of the basic formula

Proposition 3.4.5 - Let x ∈ W^{ϵ}(R).

(i) Let a ∈ k^{×} be such that neither a nor −a is a square and let K = k(\sqrt{a}), K′ = k(\sqrt{−a}). If x_{K} = 0 and x_{K′} = 0, then 2x = 0.

(ii) Let E be a finite extension of k of odd degree. If x_{E} = 0, then x = 0. Proof of (i). Let s : K → k be the trace map divided by 2. The corresponding quadratic form s_{∗}(1) is the form X^{2} + aY^{2}, i.e. < 1 > + < a > using standard notation. Similarly, there is a linear map s′ : K′ → k such that s′_{∗}(1) = < 1 > + <−a>; we have

s_{∗}(1) + s′_{∗}(1) = <1> + <1> + <a> + <−a> = <1> + <1> = 2,

because < a > + < −a > = 0 in W(k). By proposition 3.4.4, the relations x_{K} = 0 and x_{K′} = 0 imply that s_{∗}(1)x = 0 and s′_{∗}(1)x = 0; by adding up, we find 2x = 0. Proof of (ii). Since [E : k] is odd, a well-known construction of Scharlau (see e.g. [S 85, chap.2, lemma 5.8]) shows that there exists a linear form s : E → k such that s_{∗}(1) = 1 in W(k). The relation x_{E} = 0 implies s_{∗}(1)x = 0, i.e. x = 0. Remark. Part (ii) of proposition 3.4.5 is equivalent to the main result of [BFL 90].

3.5. Second reduction: from formally real to real closed

Proposition 3.5.1 - Let Ω be an algebraic closure of k. Let x ∈ W^{ϵ}(R). Suppose that x_{K} = 0 for every real closed extension K of k contained in Ω. Then x is a 2-primary torsion element, i.e. there exists r ∈ N such that 2^{r}x = 0. [Recall that a field is real closed if it is formally real, and if no proper algebraic extension of it has this property; such a field is called ordonné maximal in [A6, §2].]

Proof. Suppose that no such r exists. By Zorn’s lemma, we may choose an extension L ⊂ Ω such that x_{L} is not a 2–primary torsion element, and such that L is maximal for this property. Then prop.3.2.1 implies that L is formally real. Let us show that L is real closed, which will give a contradiction. To do so it is enough to prove (cf. [A6, §2, th.3]) that L has the following properties:

(i) Every element a of L is either a square or minus a square.

(ii) The field L does not have any non trivial odd degree extension.

Let us prove (i). If not, there would exist a ∈ L such that neither a nor −a is a square. The field extensions L(\sqrt{a})/L and L(\sqrt{−a})/L are non trivial. Because of the maximality of L, this implies that x_{L(\sqrt{a})} and x_{L(\sqrt{−a})} are 2– primary torsion elements. By part (i) of proposition 3.4.5 the same is true for x_{L}, which is a contradiction.

The proof of (ii) is similar: one uses part (ii) of proposition 3.4.5.

Corollary 3.5.2 - Let n be an odd integer. If R_{K} has property Div_{n,ϵ} for every real closed field extension K of k, then R has the same property. Proof. Let x ∈ W^{ϵ}(R) be such that nx = 0. If K is any real closed extension of k, we have x_{K} = 0, because of Div_{n,ϵ}. By proposition 3.5.1 this implies that 2^{r}x = 0 for a suitable r ∈ Z. Since nx = 0 and n is odd, we have x = 0.

Corollary 3.5.2 shows that it is enough to prove theorem 3.1.1 when k is real closed. We shall actually prove more: Theorem 3.5.3 - If k is real closed, then R has property Div_{n,ϵ} for every n > 0 and every ϵ = ±1.

In the next section, we shall check this when R is a division algebra. The general case will be done in §3.11.

3.6. The case where k is real closed and R is a division algebra Theorem 3.6.1 - Suppose that k is real closed and that R is a division algebra. Then WGr^{ϵ}(R) is isomorphic to either Z or Z ⊕ Z.

In particular, WGr^{ϵ}(R) is torsion-free. By proposition 2.6.3 this implies: Corollary 3.6.2 - Property Div_{n,ϵ} is true for every n > 0 and every ϵ = ±1. Proof of theorem 3.6.1. Up to isomorphism, R is either k, k(i) or the standard quaternion algebra over k. There are ten different possibilities, according to the choice of R, of its involution, and of ϵ :

3.6.3 - R = k with trivial involution and ϵ = 1. Then WGr^{ϵ}(R) is the Witt-Grothendieck group of quadratic forms over k. Such a form is well-defined by its signature, i.e. by the number of positive and negative coefficients when written as Σa_{i}x_{i}^{2}. Hence WGr^{ϵ}(R) ≃ Z ⊕ Z.

3.6.4 - R = k with trivial involution and ϵ = −1. The classification of alternating forms shows that such a form is well-defined by its rank, which is an even number. Hence WGr^{ϵ}(R) ≃ Z.

3.6.5 - R = k(i) with trivial involution and ϵ = 1. Here WGr^{ϵ}(R) is the same as the Witt-Grothendick group of quadratic forms over k(i); such a form is well-defined by its rank, hence WGr^{ϵ}(R) ≃ Z.

3.6.6 - R = k(i) with trivial involution and ϵ = −1. The result is the same as in case 3.6.4, i.e. WGr^{ϵ}(R) ≃ Z.

3.6.7 - R = k(i) with the involution i ↦ −i and ϵ = 1. The result is the same as in case 3.6.3, i.e. WGr^{ϵ}(R) ≃ Z ⊕ Z.

3.6.8 - R = k(i) with the involution i ↦ −i and ϵ = −1. If h is an ϵhermitian form, then i.h is (−ϵ)-hermitian. We are thus reduced to case 3.6.7, hence WGr^{ϵ}(R) ≃ Z ⊕ Z.

3.6.9 - R = quaternion algebra with basis <1, i, j, ij>, the involution being the standard one {i, j, ij ↦ −i, −j, −ij}, and ϵ = 1. The result is the same as in cases 3.6.3 and 3.6.7, namely WGr^{ϵ}(R) ≃ Z ⊕ Z.

3.6.10 - R = quaternion algebra with the standard involution and ϵ = −1. Up to isomorphism, there is only one form of rank 1, namely (x, y) ↦ xi\overline{y}. Hence WGr^{ϵ}(R) ≃ Z.

3.6.11 - R = quaternion algebra with an orthogonal involution, for instance {i, j, ij ↦ −i, j, ij}, and ϵ = 1. If h is an ϵ-hermitian form, then hi is (−ϵ)-hermitian for the standard involution of R. By 3.6.10, we thus have WGr^{ϵ}(R) ≃ Z.

3.6.12 - R = quaternion algebra with an orthogonal involution as above and ϵ = −1. The same argument as in 3.6.11 shows that WGr^{ϵ}(R) ≃ Z ⊕ Z.

This completes the proof of theorem 3.6.1. Remark. The corresponding Witt groups W^{ϵ}(R) are easily computed. They are respectively: Z, 0, Z/2Z, 0, Z, Z, Z, Z/2Z, Z/2Z, Z.

3.7. Hermitian elements

Before completing the proof of theorems 3.1.1 and 3.5.3, we recall a few basic facts on hermitian elements.

Let E be a ring with an involution σ : E → E ^{1}. Let ϵ = ±1, and put

E^{ϵ} = {z ∈ E^{×} | σ(z) = ϵz}.

If z ∈ E^{ϵ}, the map h_{z} : E × E → E defined by h_{z}(x, y) = x.z.σ(y) is an ϵ–hermitian space over E; conversely, every ϵ–hermitian space over E with underlying module E is isomorphic to h_{z} for some z ∈ E^{ϵ}.

Define an equivalence relation on E^{ϵ} by setting z ≡ z′ if there exists e ∈ E^{×} with z′ = σ(e)ze; this is equivalent to (E, h_{z}) ≃ (E, h_{z′}). Let H^{ϵ}(E, σ) be the quotient of E^{ϵ} by this equivalence relation. If z ∈ E^{ϵ}, we denote by [z] its class in H^{ϵ}(E, σ).

Lemma 3.7.1 - Let A be a ring, let A^{op} be the opposite ring, and let E = A×A^{op}. Let σ be the involution of E defined by (a, b) ↦ (b, a). Then H^{ϵ}(E, σ) = {1}. Proof. Every element of E^{ϵ} is of the form (a, ϵa) with a ∈ A^{×}. If (b, ϵb) is another such element, we have (a^{−1}, b)(a, ϵa)(b, a^{−1}) = (b, ϵb), hence (a, ϵa) ≡ (b, ϵb).

Lemma 3.7.2 - Let (A, τ) and (B, τ′) be two rings with involution, and let (E, σ) be their direct product. Then H^{ϵ}(E, σ) = H^{ϵ}(A, τ) × H^{ϵ}(B, τ′). Proof. Let (a, b), (a′, b′) ∈ E^{ϵ}. Then [(a, b)] = [(a′, b′)] in H^{ϵ}(E, σ) if and only if there exists (e, f) ∈ E^{×} such that σ(e, f)(a, b)(e, f) = (a′, b′). This is equivalent to τ(e)ae = a′ and τ′(f)bf = b′, hence to [a] = [a′] ∈ H^{ϵ}(A, τ) and [b] = [b′] ∈ H^{ϵ}(B, τ′). The lemma follows.

Lemma 3.7.3 - Let J be a two–sided ideal of E such that σ(J) = J. Suppose that J is nilpotent, and that 2 is invertible in E. Set \overline{E} = E/J and denote by \overline{σ} the involution of \overline{E} defined by σ. Then the projection map π : E → \overline{E} induces a bijection H^{ϵ}(E, σ) → H^{ϵ}(\overline{E}, \overline{σ}).

1In the next section, E will not be R, but rather the endomorphism ring of an hermitian R-space; in Bourbaki’s parlance ([A8, §1.3]), we switch from a module to its “contre-module”; from a “category with duality” point of view, we switch from an object to its endomorphism ring.

Proof. Let us first show the surjectivity of the projection E^{ϵ} → \overline{E}^{ϵ} and hence that of H^{ϵ}(E, σ) → H^{ϵ}(\overline{E}, \overline{σ}). Let u be an element of \overline{E}^{ϵ}. Choose v ∈ E with π(v) = u and let w = \frac{1}{2}(v+ϵσ(v)). We have σ(w) = ϵw and π(w) = u. Since the kernel J of π is nilpotent, and u is invertible, the relation π(w) = u implies that w is invertible. Hence we have w ∈ E^{ϵ}; this proves the surjectivity of E^{ϵ} → \overline{E}^{ϵ}.

Let us now prove the injectivity of H^{ϵ}(E, σ) → H^{ϵ}(\overline{E}, \overline{σ}). By induction, we may assume that J^{2} = 0. Let z, z′ ∈ E^{ϵ} such that [z] = [z′] in H^{ϵ}(\overline{E}, σ). Then there exists a ∈ E^{×} such that σ(a)za − z′ ∈ J. Replacing z by σ(a)za, we may assume that z′ = z + r with r ∈ J. Then we have σ(r) = ϵr. Set b = \frac{1}{2}z^{−1}r. Then zb = \frac{1}{2}r, hence σ(b)z = \frac{1}{2}r; by adding up, this gives r = zb + σ(b)z. Since J^{2} = 0, we have σ(b)zb = 0, hence z′ = σ(1 + b)z(1 + b); therefore [z] = [z′] in H^{ϵ}(E, σ), as claimed.

3.8. Classifying hermitian spaces via hermitian elements

Let ϵ_{0} = ±1, and let (M, h_{0}) be an ϵ_{0}–hermitian space over R. Set E_{M} = End(M). Let τ : E_{M} → E_{M} be the involution of E_{M} induced by h_{0}, i.e.

τ(e) = h_{0}^{−1}e^{∗}h_{0}, for e ∈ E_{M},

where e^{∗} is the adjoint of e, cf. §2.1. If (M, h) is an ϵ–hermitian space (with the same underlying module M), we have

τ(h_{0}^{−1}h) = h_{0}^{−1}(h_{0}^{−1}h)^{∗}h_{0} = h_{0}^{−1}h^{∗}(h_{0}^{−1})^{∗}h_{0} = ϵ_{0}ϵh_{0}^{−1}h. Hence h_{0}^{−1}h is an ϵ_{0}ϵ–hermitian element of (E_{M}, τ); let [h_{0}^{−1}h] be its class in H^{ϵ_{0}ϵ}(E_{M}, τ).

Lemma 3.8.1 - Sending an ϵ–hermitian space (M, h) to the element [h_{0}^{−1}h] of H^{ϵ_{0}ϵ}(E_{M}, τ) induces a bijection between the set of isomorphism classes of ϵ–hermitian spaces (M, h) and the set H^{ϵ_{0}ϵ}(E_{M}, τ). Proof. Let (M, h) and (M, h′) be two ϵ–hermitian spaces, and set u = h_{0}^{−1}h, u′ = h_{0}^{−1}h′. We have [u] = [u′] if and only if there exists e ∈ E_{M}^{×} such that u′ = τ(e)ue, i.e.

h_{0}^{−1}h′ = h_{0}^{−1}e^{∗}h_{0}.h_{0}^{−1}h.e which is equivalent to h′ = e^{∗}he and means that (M, h) and (M, h′) are isomorphic.

3.9. A reformulation of the n-division property

Let (M, h_{0}) be an ϵ_{0}–hermitian space, let E = End(M) and let σ : E → E be the involution induced by h_{0}, cf. §3.8. If n ∈ N, let E_{n} = End(M^{n}) = M_{n}(E), and let σ_{n} : E_{n} → E_{n} be defined by σ_{n}(a_{i,j}) = (σ(a_{j,i})).

Proposition 3.9.1 - Let n be an integer > 0. The following properties are equivalent :

(i) The ϵ-hermitian spaces (M, h) with underlying module M have property Div_{n} of §3.1. (ii) The map f_{n} : H^{ϵ_{0}ϵ}(E, σ) → H^{ϵ_{0}ϵ}(E_{n}, σ_{n}) given by

[u] ↦ [u_{n}] = [( \begin{matrix}u & 0 & … & 0 \\ … & … & … & … \\ 0 & … & 0 & u \end{matrix} )]

is injective, where u_{n} is the n × n–matrix with diagonal entries equal to u. [For the definition of H^{±1}(E, σ), see §3.7.]

Proof. Let us prove that (i) implies (ii). Let u, u′ ∈ E be such that f_{n}([u]) = f_{n}([u′]). Let (M, h), respectively (M, h′), be ϵ–hermitian forms such that the isomorphism class of (M, h) corresponds to [u] ∈ H^{ϵ_{0}ϵ}(E, σ), and that the isomorphism class of (M, h′) corresponds to [u′] ∈ H^{ϵ_{0}ϵ}(E, σ). Then the isomorphism class of n • (M, h) corresponds to f_{n}([u]), and the isomorphism class of n • (M, h′) corresponds to f_{n}([u′]). By hypothesis, we have f_{n}([u]) = f_{n}([u′]), hence n • (M, h) ≃ n • (M, h′). By (i), we have (M, h) ≃ (M, h′), thus [u] = [u′] in H^{ϵ_{0}ϵ}(E, σ). This implies that f_{n} is injective, hence (ii) holds.

Conversely, let us prove that (ii) implies (i). Let (M, h) and (M, h′) be two ϵ–hermitian spaces, and let u, u′ ∈ E such that [u], respectively [u′], are the elements of H^{ϵ_{0}ϵ}(E, σ) corresponding to the isomorphism classes of (M, h), respectively (M, h′), by the bijection of lemma 3.8.1. Then the isomorphism class of n•(M, h) corresponds to f_{n}([u]), and the isomorphism class of n•(M, h′) corresponds to f_{n}([u′])). By hypothesis, we have n • (M, h) ≃ n • (M, h′), hence f_{n}(u) = f_{n}(u′). By (ii), the map f_{n} is injective, therefore we have (M, h) ≃ (M, h′), and this implies (i).

3.10. An injectivity property

Let E be a finite dimensional k–algebra, and let σ : E → E be a k–linear involution. If n ∈ N, let E_{n} = M_{n}(E), and let σ_{n} : E_{n} → E_{n} be defined by σ_{n}(a_{i,j}) = (σ(a_{j,i})).

Theorem 3.10.1 - Suppose that k is real closed and that n is > 0. Then the map f_{n} : H^{ϵ}(E, σ) → H^{ϵ}(E_{n}, σ_{n}) given by

[u] ↦ [u_{n}] = [( \begin{matrix}u & 0 & … & 0 \\ … & … & … & … \\ 0 & … & 0 & u \end{matrix} )]

is injective, where u_{n} is the n × n–matrix with diagonal entries equal to u.

Proof. The case of a simple algebra.

Suppose first that E is a simple k–algebra. Then E ≃ M_{m}(D) for some m ∈ N and some division algebra D. Let M = D^{m}. Then E = End_{D}(M). It is well-known (see e.g. [S 85, chap.8, cor.8.3] or [KMRT 98, chap.I, th.3.1]) that there exists a k-linear involution of D of the same kind as σ (i.e. it is equal to σ on the center of D). Let ρ : D → D be such an involution. By [KMRT 98, chap.I, th.4.2] there exist ϵ_{0} = ±1 and a nonsingular ϵ_{0}-form h_{0} on M such that the involution σ of E is induced by h_{0} in the sense defined in §3.8.

Let u, u′ ∈ E^{ϵ} such that f_{n}([u]) = f_{n}([u′]), and let (M, h), (M, h′) be ϵ_{0}ϵ– hermitian spaces over D such that [u], [u′] ∈ H^{ϵ}(E, σ) correspond to the isomorphism classes of h, respectively h′, by the bijection of lemma 3.9.1. The elements [u_{n}], [u′_{n}] ∈ H^{ϵ}(E_{n}, σ_{n}) correspond to the isomorphism classes of n • h, respectively n • h′. We have [u_{n}] = f([u]) = f([u′]) = [u′_{n}], hence n • h ≃ n • h′. Since k is real closed, corollary 3.6.2, applied to D instead of R, shows that h ≃ h′. Therefore [u] = [u′], and this proves the injectivity of f_{n} in the case where E is a simple algebra. The case of a semi–simple algebra.

Suppose now that E is semi–simple. Then

E ≃ E_{1} × … × E_{r} × A × A^{op},

where E_{1}, …, E_{r} are simple algebras which are stable under the involution σ, and where the restriction of σ to A × A^{op} exchanges the two factors. Applying lemma 3.7.1 and lemma 3.7.2 we are reduced to the case where E is a simple algebra, and we already know that the result is true in this case. General case.

Let \overline{E} = E/rad(E). Then \overline{E} is semi–simple, and σ induces a k–linear involution \overline{σ} : \overline{E} → \overline{E}. Set \overline{E}_{n} = E_{n}/rad(E_{n}). Then \overline{E}_{n} = M_{n}(\overline{E}), and σ_{n} induces \overline{σ}_{n} : \overline{E}_{n} → \overline{E}_{n}. We have the following commutative diagram

H^{ϵ}(E, σ) ⟶^{f_{n}} H^{ϵ}(E_{n}, σ)
↓  ↓
H^{ϵ}(\overline{E}, \overline{σ}) ⟶^{\overline{f}_{n}} H^{ϵ}(\overline{E}_{n}, \overline{σ}),

where the vertical maps are induced by the projection E → \overline{E}. By lemma 3.7.3, these maps are bijective. As \overline{E} is semi–simple, the map \overline{f}_{n} is injective, hence f_{n} is also injective. This concludes the proof.

3.11. Proofs of theorem 3.5.3 and theorem 3.1.1

As we have already seen at the end of §3.5, it is enough to prove theorem 3.5.3. This means proving that, if k is real closed, and if n > 0, then every ϵhermitian space (M, h) over R has property Div_{n}. Let (M′, h′) be an ϵ-hermitian space over R such that n • (M, h) ≃ n • (M′, h′). As noticed in §3.1, we may assume that M′ = M. Let E = End(M) and let τ be the involution of E induced by h, cf. §3.8. As explained in §3.8 (applied to h_{0} = h), the hermitian forms h and h′ define ϵ-hermitian elements u = h_{0}^{−1}h = 1 and u′ = h_{0}^{−1}h′ of (E, τ); let [u] and [u′] be their classes in H^{ϵ}(E, τ). Since n • (M, h) ≃ n • (M, h′), we have [u_{n}] = [u′_{n}], where u_{n} and u′_{n} are defined as in proposition 3.9.1 and theorem 3.10.1. By theorem 3.10.1 this implies [u] = [u′], hence (M, h) ≃ (M, h′) by proposition 3.8.1. This concludes the proof. Question. In several of the proofs above, we have made an essential use of the existence of a commutative subfield k of R having the following two properties:

a) The involution of R is k-linear.

b) dim_{k}R < ∞.

We do not know what happens when there is no such k. For instance, suppose that R is a skew field with involution which is of infinite dimension over its center, and that n is odd; does Div_{n} hold for the R-hermitian spaces of finite dimension over R ? It does^{2} when the characteristic is ≠ 0 But what about the characteristic 0 case ?

§4. Induction and restriction for G-quadratic forms

Let k be a field of characteristic ≠ 2. Let G be a finite group and let S be a 2-Sylow subgroup of G. Our aim is to prove th.1.1.2. We recall its statement:

Theorem 1.1.2 - Let (V_{1}, q_{1}) and (V_{2}, q_{2}) be two S-quadratic spaces over k. Suppose that:

Res_{S}^{G} Ind_{S}^{G}(V_{1}, q_{1}) ≃_{S} Res_{S}^{G} Ind_{S}^{G}(V_{2}, q_{2}).

Then:

Ind_{S}^{G}(V_{1}, q_{1}) ≃_{G} Ind_{S}^{G}(V_{2}, q_{2}).

The proof is based on:

• the Witt-type cancellation theorem, cf. §2;

• the “no odd torsion” principle, cf. §3;

• the elementary properties of the Burnside rings, cf. §4.2 below.

4.1. Algebraic properties of the projection formula

Let A and B be two commutative rings, and let r : B → A and i : A → B be respectively a ring homomorphism and an additive homomorphism such that:

(4.1.1) i(r(y)x) = y.i(x) for every x ∈ A and y ∈ B. [This formula means that i : A → B is B-linear if A is viewed as a B-module via r. In K-theory, such a formula is often called a “projection formula”, and i and r are denoted by symbols such as f_{∗} and f^{∗}. In §4.3 below, i will be Ind_{S}^{G} and r will be Res_{S}^{G}.]

We put

(4.1.2) Q = i(1_{A}) ∈ B and q = r(Q) ∈ A, where 1_{A} denotes the unit element of the ring A.

Formula (4.1.1), applied to x = 1_{A}, gives

(4.1.3) i(r(y)) = Q.y for all y ∈ B.

Let us define R : A → A by:

(4.1.4) R = r ∘ i.

2Use the fact that, over F_{p}, p ≠ 2, a quadratic form < 1, ..., 1 > of odd rank is isomorphic to the direct sum of a hyperbolic form and either <1> or <−1>.

It is an additive endomorphism of A; we want to compare its kernel to that of i. To do so, let (F, n) be a pair where F ∈ Z[t] is a one-variable polynomial with coefficients in Z and n is an integer; assume that:

(4.1.5) q.F (q) = n.1_{A}.

Proposition 4.1.2. - If A, B, i, r, F, n are as above, we have :

(4.1.6) n.i(a) = i(F (q).R(a)) for every a ∈ A.

Proof. Let q′ = F (q) and let Q′ = F (Q). We have: i(q′.R(a)) = Q′.i(R(a)) by (4.1.1) applied to x = R(a) and y = Q′

= Q′Q.i(a) by (4.1.3) applied to y = i(a)

= i(r(Q′Q)a) by (4.1.1) applied to x = a and y = Q′Q
= i(n.a) since r(Q′Q) = q′q = n.1_{A}
= n.i(a).

Corollary 4.1.3 - Assume that B is n-torsion-free. Then Ker(i) = Ker(R). Proof. The inclusion Ker(i) ⊂ Ker(R) is obvious, and the opposite inclusion follows from (4.1.6), which shows that n.i(a) = 0 if R(a) = 0. Remark. Corollary 4.1.3 is all we need for the proof of theorem 1.1.2. However, the following result is worth mentioning: Proposition 4.1.4 - (1) The kernel and the cokernel of the natural map

Ker(R) ⊕ Im(R) → A

are killed by n.

(2) The same is true for the map Ker(r) ⊕ Im(i) → B. [Recall that an abelian group X is killed by n if n.x = 0 for every x ∈ X.]

Proof of (1). We have

(4.1.7) R^{2}(a) = q.R(a) for every a ∈ A. Indeed, by (4.1.3) we have i(r(i(a))) = Q.i(a), hence

R^{2}(a) = r(i(r(i(a)))) = r(Q.i(a)) = r(Q).r(i(a)) = q.R(a).

Let now x be an element of Ker(R) ∩ Im(R), and let us write x as R(a), with a ∈ A. By (4.1.7), we have q.R(a) = 0, i.e. q.x = 0, hence n.x = 0 by (4.1.5). This shows that the kernel of Ker(R) ⊕ Im(R) → A is killed by n. In order to prove the same result for the cokernel, we need to show that, for every x ∈ A, there exists y ∈ Im(R) and z ∈ Ker(R) with nx = y + z. We choose y = F (q).R(x) = r(F (Q).i(x)) = R(F (q).x), and we define z as z = n.x − y. It remains to check that R(z) = 0, i.e. that n.R(x) = R(y); this follows from R(y) = R^{2}(F (q).x) = q.R(F (q).x) = q.F (q).R(x) = n.R(x).

Proof of (2). If b ∈ B belongs to Ker(r) ∩ Im(i), we may write b as i(a) with a ∈ A, and we have r(i(a)) = 0, i.e. R(a) = 0, which implies n.i(a) = 0 by (4.1.6), i.e. n.b = 0. If b is any element of B, put y = Q.F (Q).b and z = n.b−y. We have r(y) = q.F (q).r(b) = n.r(b), hence r(z) = 0. Since y belongs to Im(i) by (4.1.3), the relation n.b = y + z shows that n.b belongs to Ker(r) ⊕ Im(i).

Remark. The above results are only interesting when the integer n can be chosen such that the rings A and B have no n-torsion. In the case we need, it will be enough that n be odd; this is what the Burnside ring method is going to give us.

4.2. The Burnside ring

Definitions and Notation

If G is a finite group, we denote by Burn(G) its Burnside ring, i.e. the Grothendieck group of the category of finite G-sets, cf. [B 91, §5.4]. Every finite G-set X defines an element b_{X} of Burn(G), and we have:

(4.2.1) b_{X⊔Y} = b_{X} + b_{Y} and

(4.2.2) b_{X×Y} = b_{X}.b_{Y} . Let H be a set of representatives of the subgroups of G up to G-conjugation. The family {b_{G/H}}_{H∈H} is a Z-basis of Burn(G); hence Burn(G) is a free Z-module of rank h = |H|. If H is a subgroup of G, there is a unique ring homomorphism

(4.2.3) f_{H} : Burn(G) → Z such that f_{H}(b_{X}) = number of points of X which are fixed by H.

The family (f_{H})_{H∈H} gives a ring homomorphism

(4.2.4) f : Burn(G) → Z^{H}, where Z^{H} denotes the product of h copies of Z indexed by H. This homomorphism is injective, and its cokernel is finite; we may thus view Z^{H} as the normalization of the ring Burn(G). [As shown by A. Dress, this leads to an explicit description of Spec Burn(G), as a union of h copies of Spec Z with suitable glueing; the precise statement is given in [B 91, th.5.4.6] where however f_{K}(x) is misprinted as f_{H}(x). A remarkable corollary is that Spec Burn(G) is connected if and only if G is solvable, cf. [B 91, cor.5.4.8].]

Characteristic polynomial and norm

Let x be an element of Burn(G). The multiplication by x is an endomorphism of the additive group Burn(G) ≃ Z^{h}. Let P_{x}(t) ∈ Z[t] be its characteristic polynomial; it is a monic polynomial of degree h. By the Cayley-Hamilton formula, we have

(4.2.5) P_{x}(x) = 0 in Burn(G).

The embedding (4.2.4) shows that P_{x}(t) is also the characteristic polynomial of x acting by multiplication on Z^{H}, hence :

(4.2.6) P_{x}(t) = ∏_{H∈H}(t − f_{H}(x)). The norm N(x) of x is

(4.2.7) N(x) = ∏_{H∈H} f_{H}(x).

It is the constant term of (−1)^{h}P_{x}(t).

Proposition 4.2.1 - Let p be a prime number, and suppose that G is a p-group. Let X be a finite G-set and let x = b_{X} be the corresponding element of Burn(G). Assume that |X| = CardX is not divisible by p. Then :

(i) N(x) is not divisible by p.

(ii) There exists F_{x} ∈ Z[t] such that x.F_{x}(x) = N(x) in Burn(G).

Proof. Let H be a subgroup of G and let X^{H} be the set of all elements of X which are fixed by H. Since H is a p-group, we have |X^{H}| ≡ |X| (mod p), hence f_{H}(x) = |X^{H}| is not divisible by p; by (4.2.7), the same is true for N(x). This proves (i). Since N(x) is the constant term of (−1)^{h}P_{x}(t), we may define a polynomial F_{x}(t) by the formula

t.F_{x}(t) = N(x) − (−1)^{h}P_{x}(t),

and we have x.F_{x}(x) = N(x) − (−1)^{h}P_{x}(x) = N(x), cf. (4.2.5). This proves (ii).

4.3. Proof of theorem 1.1.2

Let Λ(G) denote the Grothendieck-Witt group of the G-quadratic spaces over k, i.e. the group denoted by WGr^{1}(k[G]) in §2.6. The tensor product operation makes Λ(G) a commutative ring; this reflects the fact that k[G] is a bialgebra (“bigèbre” in French), and not merely an algebra, cf. [A8, §11.7].

The ring Λ(S) is defined in a similar way. The induction and restriction functors give rise to maps

Ind_{S}^{G} : Λ(S) → Λ(G) and Res_{S}^{G} : Λ(G) → Λ(S). The rings A = Λ(S) and B = Λ(G), together with the maps i = Ind_{S}^{G} and r = Res_{S}^{G}, have the properties described at the beginning of §4.1; in particular, formula (4.1.1) is valid, as a simple computation shows. We may then apply the constructions of §4.1, and define:

Q = i(1_{A}) ∈ B and q = r(Q) ∈ A.

Lemma 4.3.1 - There exists F ∈ Z[t] and n ∈ Z, with n odd, such that q.F (q) = n.1_{A}, as in (4.1.5). Proof. Let Burn(G) be the Burnside ring of the group G, cf §4.2. There is a unique ring homomorphism

ε_{G} : Burn(G) → B = Λ(G), having the following property: if X is any finite G-set, ε_{G}(b_{X}) is equal to [q_{X}] , where q_{X} is the unique G-quadratic form with an orthonormal basis isomorphic to X as a G-set.

Let now X = G/S, with the natural action of G. The corresponding Gquadratic form is Ind_{S}^{G}(1) , where 1 means the unit quadratic form of rank 1, with trivial action of S; its class in B is Q = i(1_{A}).

We may apply the same construction to S; we have ε_{S} : Burn(S) → A = Λ(S), and ε_{S}(b_{X}) = r(Q) = q. By proposition 4.2.1, applied to S, to X = G/S, and to p = 2, there exist F ∈ Z[t] and n ∈ Z, with n odd, such that b_{X}.F (b_{X}) = n in Burn(S). Applying ε_{S} to this equality gives q.F (q) = n.1_{A} in A, as wanted.

Remark. Using the Burnside ring in questions involving G-functors is a very efficient technique, which was introduced by A. Dress in the early 1970s. See e.g. [B 91, §5.6].

Theorem 4.3.2 - The maps

Res_{S}^{G} ∘ Ind_{S}^{G} : Λ(S) → Λ(S) and Ind_{S}^{G} : Λ(S) → Λ(G) have the same kernel. Proof. Let n be as in lemma 4.3.1. Since n is odd, the ring B = Λ(G) is ntorsion-free (see theorem 3.1.1). By corollary 4.1.3, we have Ker(i) = Ker(r ∘ i).

End of the proof of theorem 1.1.2.

Let v_{1} and v_{2} be the classes of (V_{1}, q_{1}) and (V_{2}, q_{2}) in Λ(S). By assumption, v_{1} − v_{2} belongs to the kernel of Res_{S}^{G} ∘ Ind_{S}^{G}; by theorem 4.3.2, it belongs to the kernel of Ind_{S}^{G}. By formula (2.6.1), this means that Ind_{S}^{G}(V_{1}, q_{1}) is G–isomorphic to Ind_{S}^{G}(V_{2}, q_{2}).

References

[A6] N. Bourbaki, Algèbre, Chap.6, Groupes et corps ordonnés, Masson, Paris, 1981; new printing, Springer–Verlag, 2006; English translation, Springer-Verlag, 1998.

[A8] ——– , Algèbre, Chap.8, Modules et anneaux semi–simples, new revised edition, Springer–Verlag, 2012.

[A9] ——– , Algèbre, Chap.9, Formes sesquilinéaires et formes quadratiques, Hermann, Paris, 1959; new printing, Springer-Verlag, 2006.

[BFL 90] E. Bayer–Fluckiger & H.W. Lenstra, Jr., Forms in odd degree extensions and self-dual normal bases, Amer. J. Math. 112 (1990), 359-373.

[BFP 11] E. Bayer–Fluckiger & R. Parimala, Galois Algebras, Hasse Principle and Induction–Restriction Methods, Documenta Math. 16 (2011), 677–707; Errata, to appear.

[BFS 94] E. Bayer–Fluckiger & J–P. Serre, Torsions quadratiques et bases normales autoduales, Amer. J. Math. 116 (1994), 1–64.

[B 91] D.J. Benson, Representations and Cohomology I, Cambridge studies in advanced mathematics 30, Cambridge University Press, Cambridge, 1991.

[K 91] M. Knus, Quadratic and Hermitian Forms over Rings, Grundlehren math. Wiss. 294, Springer-Verlag, 1991.

[KMRT 98] M. Knus, A. Merkurjev, M. Rost & J–P. Tignol, The Book of Involutions, AMS Colloquium Publications 44, 1998.

[La 04] T.Y. Lam, Introduction to Quadratic Forms over Fields, AMS, GSM 67, 2004.

[M 86] J. Morales, Integral bilinear forms with a group action, J. Algebra 98 (1986), 470-484.

[QSS 79] H–G. Quebbemann, W. Scharlau & M. Schulte, Quadratic and hermitian forms in additive and abelian categories, J. Algebra 59 (1979), 264– 289.

[QSSS 76] H–G. Quebbemann, R. Scharlau, W. Scharlau & M. Schulte, Quadratische Formen in additiven Kategorien, Bull. Soc. Math. France, Mémoire 48 (1976), 93–101.

[R 11] C.R. Riehm, Introduction to Orthogonal, Symplectic and Unitary Representations of Finite Groups, Fields Institute Monographies 28, AMS, 2011.

[S 70] W. Scharlau, Induction theorems and the structure of the Witt group, Invent. math. 11 (1970), 37–44.

[S 85] ——– , Quadratic and Hermitian Forms, Grundlehren math. Wiss. 270, Springer–Verlag, 1985.

E. Bayer–Fluckiger École Polytechnique Fédérale de Lausanne EPFL/FSB/MATHGEOM/CSAG Station 8 1015 Lausanne, Switzerland eva.bayer@epfl.ch

R. Parimala Department of Mathematics & Computer Science Emory University Atlanta, GA 30322, USA parimala@mathcs.emory.edu

J–P. Serre Collège de France 3 rue d’Ulm 75005 PARIS, France serre@noos.fr