Sesquilinear forms over rings with involution

Eva Bayer-Fluckiger and Daniel Arnold Moldovan^{∗} École Polytechnique Fédérale de Lausanne, Switzerland

December 29, 2012

Abstract Many classical results concerning quadratic forms have been extended to hermitian forms over algebras with involution. However, not much is known in the case of sesquilinear forms without any symmetry property. The present paper will establish a Witt cancellation result, an analogue of Springer’s theorem, as well as some local-global and finiteness results in this context.

Mathematics Subject Classification (2000) 11E39, 11E81.

Keywords sesquilinear forms, hermitian forms, hermitian categories.

Introduction

Many classical results on quadratic forms have been extended to hermitian forms over some rings with involution (see for instance [8], [13]). However, not much is known about sesquilinear forms without any symmetry property. The aim of this paper is to show that the category of sesquilinear forms over rings with involution is equivalent to the category of unimodular hermitian forms over a suitable additive category (cf. §1 – §4), and then to apply the theory of Quebbemann, Scharlau, Scharlau and Schulte (cf. [10]–[11]) as well as known results concerning hermitian forms.

This enables us to prove Witt’s cancellation theorem (see theorem 6.1) and an analogue of Springer’s theorem (see theorem 7.1) for sesquilinear forms over finitedimensional algebras with involution over fields of characteristic different from 2. We also obtain local-global results (see theorem 9.1 and its corollary) as well as finiteness theorems.

∗Partially supported by the Swiss National Science Fundation, grant 200020-109174/1

§1. Sesquilinear forms over rings with involution

Let A be a ring. An involution on A is by definition an additive map σ : A → A such that σ(ab) = σ(b)σ(a) for all a, b ∈ A and σ^{2} is the identity. Let V be a right A–module of finite type. A sesquilinear form over (A, σ) is a biadditive map s : V × V → A satisfying the condition s(ax, by) = σ(a)s(x, y)b for all x, y ∈ V and all a, b ∈ A. The orthogonal sum of two sesquilinear forms (V, s) and (V′, s′) is by definition the form (V ⊕ V′, s ⊕ s′) defined by

(s ⊕ s′)(x ⊕ x′, y ⊕ y′) = s(x, y) + s′(x′, y′)

for all x, y ∈ V and x′, y′ ∈ V′. Two sesquilinear forms (V, s) and (V′, s′) are called isometric if there exists an isomorphism of A-modules f : V ˜→V′ such that s′(f(x), f(y)) = s(x, y) for all x, y ∈ V.

Let V^{∗} = Hom_{A}(V, A). Then V^{∗} has a structure of right A–module given by f(x) · a = σ(a)f(x) for all a ∈ A, f ∈ V^{∗} and a ∈ A. We say that V is reflexive if the homomorphism of right A-modules e_{V} : V → V^{∗∗} defined by e_{V} (x)(f) = σ(f(x)) for all x ∈ V and f ∈ V^{∗} is bijective.

A sesquilinear form (V, s) induces two homomorphisms of left A-modules V → V^{∗}, called its left, respectively right adjoint, namely s_{l} : V → V^{∗} defined by s_{l}(x)(y) = s(x, y) and s_{r} : V → V^{∗} given by s_{r}(x)(y) = σ(s(y, x)) for all x, y ∈ V.

Let ℛ be a category of reflexive right A-modules. The morphisms of this category are homomorphisms of A-modules.

Let us denote by 𝒮_{ℛ}(A, σ) the category of sesquilinear forms over (A, σ) defined on objects of ℛ. The morphisms of this category are isometries.

§2. Hermitian categories

The aim of this section is to recall the notions of sesquilinear and hermitian forms in additive categories as defined in [10], [11] (see also [8], [13]).

Let 𝒞 be an additive category. Let ^{∗} : 𝒞 → 𝒞 be a duality functor, i.e. an additive contravariant functor with a natural isomorphism (E_{C})_{C∈𝒞} : id → ^{∗∗} such that E_{C}^{∗}E_{C^{∗}} = id_{C^{∗}} for all C ∈ 𝒞. A hermitian form in the category 𝒞 is a pair (C, s), where C is an object of 𝒞 and s : C → C^{∗} such that s = s^{∗}E_{C}. Orthogonal sums are defined in the obvious way.

Let (C, h) and (C′, h′) be two hermitian forms in 𝒞. We say that they are isometric if there exists an isomorphism f : C ˜→C′ in the category 𝒞 such that h = f^{∗}h′f.

We denote by ℋ(𝒞) the category of unimodular hermitian forms in the category 𝒞. The morphisms are isometries.

We observe that if we take 𝒞 = ℛ then the above notion coincides with the notion of hermitian form over (A, σ) defined on objects of ℛ.

§3. An additive category

Let (A, σ) be a ring with involution and ℛ be a category of reflexive right A-modules of finite type.l. Let us consider the category ℳ_{ℛ} with objects of the form (V, W, f_{1}, f_{2}), where V and W are objects of ℛ of finite type and f_{1} : V → W, f_{2} : V → W are homomorphisms of A–modules. A morphism (V, W, f_{1}, f_{2}) → (V′, W′, f′_{1}, f′_{2}) in ℳ_{ℛ} is a pair (ϕ, ψ) of A–linear homomorphisms ϕ : V → V′ and ψ : W → W′ such that f′_{1}ϕ = ψf_{1} and f′_{2}ϕ = ψf_{2}. By defining direct sums in the obvious way we see that ℳ_{ℛ} is an additive category. It is called the category of double arrows between objects of ℛ.

Let (W^{∗}, V^{∗}, f_{2}^{∗}, f_{1}^{∗}) be the dual of (V, W, f_{1}, f_{2}) and set E_{(V,W,f_{1},f_{2})} = (e_{V} , e_{W}). This defines a duality on the category ℳ_{ℛ}. We denote by ℋ(ℳ_{ℛ}) the category of unimodular hermitian forms in the hermitian category ℳ_{ℛ}, as defined in the previous section.

§4. An equivalence of categories

Let (A, σ) be a ring with involution and ℛ be a category of reflexive right Amodules of finite type. The aim of this section is to prove that the category of sesquilinear forms 𝒮_{ℛ}(A, σ) is equivalent to the hermitian category ℋ(ℳ_{ℛ}) defined in §3. We will use the methods of [2].

We define a functor ℱ from the category 𝒮_{ℛ}(A, σ) to the category ℋ(ℳ_{ℛ}) as follows. Let (V, s) be an object of 𝒮_{ℛ}(A, σ), and let s_{ℓ} : V → V^{∗} and s_{r} : V → V^{∗} be the left, respectively the right adjoint of (V, s) (cf. §1). Then (V, V^{∗}, s_{ℓ}, s_{r}) is an object of ℳ_{ℛ} and it is easy to check that the pair (e_{V} , id_{V^{∗}}) defines a unimodular hermitian form on (V, V^{∗}, s_{ℓ}, s_{r}). Set ℱ(V, s) = ((V, V^{∗}, s_{ℓ}, s_{r}), (e_{V} , id_{V^{∗}})). For an isomorphism ϕ : (V, s) → (W, t) of sesquilinear forms, where V, W ∈ ℛ, set ℱ(ϕ) = (ϕ, ϕ^{∗−1}).

4.1. Theorem The functor ℱ is an equivalence of categories between 𝒮_{ℛ}(A, σ) and ℋ(ℳ_{ℛ}).

Proof. We already know that ℱ sends sesquilinear forms over (A, σ) defined on objects of ℛ to objects of ℋ(ℳ_{ℛ}). Let us check that ℱ sends morphisms to morphisms. Let ϕ : (V, s) → (W, t) be an isomorphism between two objects of 𝒮_{ℛ(A,σ)}. Then (ϕ, ϕ^{∗−1}) : (V, V^{∗}, s_{ℓ}, s_{r}) → (W, W^{∗}, t_{ℓ}, t_{r}) is a morphism in ℳ_{ℛ}. Moreover, as ϕ^{∗∗}e_{V} = e_{W} ϕ, it is also an isomorphism in ℋ(ℳ_{ℛ}).

Let us define a functor 𝒢 from ℋ(ℳ_{ℛ}) to 𝒮_{ℛ}(A, σ). Let (M, ζ) be an object of ℋ(ℳ_{ℛ}) with M = (V, W, f_{1}, f_{2}) and ζ = (ϕ, ψ). Let us define a sesquilinear form s : V × V → A by s(x, y) = (ψf_{2})(x)(y) for all x, y ∈ V and set 𝒢(V, W, f_{1}, f_{2}) = (V, s). Then (V, s) is an object of 𝒮_{ℛ}(A, σ). For a morphism λ = (λ_{1}, λ_{2}) between two objects of ℋ(ℳ_{ℛ}) set 𝒢(λ) = λ_{1}.

Let us check that 𝒢 sends morphisms to morphisms. Let

λ = (λ_{1}, λ_{2}) : ((V, W, f_{1}, f_{2}), (ϕ, ψ)) → ((V′, W′, f′_{1}, f′_{2}), (ϕ′, ψ′))

be a morphism between two objects of ℋ(ℳ_{ℛ}). Then λ_{1} : V → V′ and λ_{2} : W → W′ are isomorphisms such that λ_{2}f_{1} = f′_{1}λ_{1} and λ_{2}f_{2} = f′_{2}λ_{1}. In addition we have (ψ, ψ) = (λ_{1}, λ_{2})^{∗}(ϕ′, ψ′)(λ_{1}, λ_{2}), from which we deduce that ϕ = λ_{2}^{∗}ϕ′λ_{1} and ψ = λ_{1}^{∗}ψ′λ_{2}. It follows that

λ_{1}^{∗}ψ′f′_{2}λ_{1} = ψλ_{2}^{−1}f′_{2}λ_{1} = ψf_{2},

so λ_{1} is a morphism from (V, ψf_{2}) to (V′, ψ′f′_{2}).

It is straightforward to check that ℱ𝒢 is isomorphic to the identity in ℋ(ℳ_{ℛ}), and that 𝒢ℱ is isomorphic to the identity in 𝒮_{ℛ}(A, σ). Therefore ℱ : 𝒮_{ℛ}(A, σ) → ℋ(ℳ_{ℛ}) is an equivalence of categories.

§5. Sesquilinear forms corresponding to a given object of ℳ_{ℛ}

Let A be a ring with an involution σ and ℛ be a category of reflexive right Amodules of finite type such that if V is an object of ℛ then V^{∗} is an object of ℛ as well. If (M, s) is a sesquilinear form over (A, σ) defined on an object of ℛ, then we denote by q(M, s) the corresponding object (M, M^{∗}, s_{l}, s_{r}) of the category ℳ_{ℛ}. In this section we describe the set of isometry classes of sesquilinear forms over (A, σ) corresponding by theorem 4.1 to a given object of the category ℳ_{ℛ}.

Let us fix a sesquilinear form (M_{0}, s_{0}) over (A, σ), where M_{0} ∈ ℛ, and consider the unimodular hermitian form η_{M_{0}} = (e_{M_{0}}, id_{M_{0}^{∗}}) defined on Q_{0} = q(M_{0}, s_{0}). Let E be the endomorphism ring of the object Q_{0} in ℳ_{ℛ}. The form η_{M_{0}} induces an involution on E, defined by f̃ = η_{M_{0}}^{−1}f^{∗}η_{M_{0}} for all f ∈ E, where f^{∗} denotes the dual of f in ℳ_{ℛ}.

On the set E^{+} = {f ∈ E| f̃ = f} we define the following equivalence relation: f ≡ f′ if there exists a g ∈ E^{×} such that g̃fg = f′. Let H( , E^{×}) denote the set of equivalence classes. It is clear that H( , E^{×}) is in bijection with the set of isometry classes of unimodular hermitian forms of rank one over (E, ).

5.1. Theorem The set of isometry classes of sesquilinear forms (M, s) ∈ 𝒮_{ℛ}(A, σ) such that q(M, s) ≃ Q_{0} is in bijection with H( , E^{×}).

Proof. For every sesquilinear form (M, s) ∈ 𝒮_{ℛ}(A, σ) such that q(M, s) ≃ Q_{0} we consider a fixed isomorphism φ : q(M, h) ˜→Q_{0}. Through this isomorphism the unimodular hermitian form (e_{M}, id_{M^{∗}}) on q(M, s) induces a unimodular hermitian form η_{M} = φ^{∗−1}(e_{M}, id_{M^{∗}})φ^{−1} on Q_{0}. We define the following map:

{[(M, s)] | (M, s) ∈ 𝒮_{ℛ}(A, σ), q(M, h) ≃ Q_{0}} → H( , E^{×})
[(M, s)] ↦ η_{M_{0}}^{−1}η_{M}.

The above map is well defined since

(M, s) ≃ (N, t) if and only if η_{M_{0}}^{−1}η_{M} ≡ η_{M_{0}}^{−1}η_{N}.

This equivalence also shows its injectivity. Its surjectivity is easy to check.

§6. Witt’s cancellation theorem for sesquilinear forms

The aim of this section is to prove a cancellation theorem for sesquilinear forms. Let K be a field of characteristic different from 2, let A be a finite-dimensional K-algebra and let σ be an involution on A. Denote by ℛ the category of reflexive right A-modules which are finite dimensional K-vector spaces.

6.1. Theorem Let (M, s), (M_{1}, s_{1}) and (M_{2}, s_{2}) be sesquilinear forms over (A, σ) defined over objects of ℛ such that

(M_{1}, s_{1}) ⊕ (M, s) ≃ (M_{2}, s_{2}) ⊕ (M, s).

Then we have (M_{1}, s_{1}) ≃ (M_{2}, s_{2}).

By the equivalence between the categories 𝒮_{ℛ}(A, σ) and ℋ(ℳ_{ℛ}) given by theorem 4.1, it is enough to prove that Witt’s cancellation theorem holds in the category ℋ(ℳ_{ℛ}). This is the purpose of proposition 6.3. For its proof we use a result of Quebbemann, Scharlau and Schulte (see [10], 3.4.1):

6.2. Theorem Let 𝒞 be a hermitian category satisfying the following conditions:

(i) All idempotents of 𝒞 split, i.e. for any object M of 𝒞 and for any idempotent e ∈ End(M), there exist an object M′ ∈ 𝒞 and morphisms i : M′ → M, j : M → M′ such that ji = id_{M′} and ij = e.

(ii) Every object M of 𝒞 has a decomposition of the form M ≃ N_{1} ⊕...⊕N_{r} with N_{i} ∈ 𝒞 indecomposable and End(N_{i}) a local ring in which 2 is invertible.

(iii) For every object M of 𝒞 the ring End(M) is J(M)-adically complete, where J(M) is the Jacobson radical of End(M).

If (M, h), (M_{1}, h_{1}) and (M_{2}, h_{2}) are unimodular hermitian forms in 𝒞 such that

(M_{1}, h_{1}) ⊕ (M, h) ≃ (M_{2}, h_{2}) ⊕ (M, h),

then (M_{1}, h_{1}) ≃ (M_{2}, h_{2}).

This theorem has been originally formulated for quadratic forms in 𝒞. But since 2 is invertible in the endomorphism ring of every object of 𝒞, the categories of quadratic and unimodular hermitian forms in 𝒞 are isomorphic. We deduce that Witt’s cancellation holds for unimodular hermitian forms in 𝒞 as well.

6.3. Proposition Let (Q, η), (Q_{1}, η_{1}) and (Q_{2}, η_{2}) be unimodular hermitian forms in the category ℳ_{ℛ} such that

(Q_{1}, η_{1}) ⊕ (Q, η) ≃ (Q_{2}, η_{2}) ⊕ (Q, η).

It then follows that (Q_{1}, η_{1}) ≃ (Q_{2}, η_{2}).

Proof. As in [2], proposition 2, we check that the category ℳ_{ℛ} satisfies the conditions (i), (ii) and (iii) above. It follows from theorem 6.2 that Witt’s cancellation theorem is true in the category ℋ(ℳ_{ℛ}).

§7. Springer’s theorem for sesquilinear forms

The classical theorem of Springer states that if two quadratic forms over a field of characteristic different from 2 become isometric over an extension of odd degree, then they are already isometric over the base field. In this section we prove an analogue of Springer’s theorem for sesquilinear forms defined on finitedimensional algebras with involution over a field of characteristic ≠ 2.

Let K be a field of characteristic different from 2, A be a finite-dimensional Kalgebra and σ be a K-linear involution. We also consider a finite extension L of K, the finite-dimensional L-algebra A_{L} = A ⊗_{K} L and the L-linear involution σ_{L} = σ ⊗ id_{L} on A_{L}. If (M, s) is a sesquilinear form over (A, σ), then we denote by (M_{L}, s_{L}) the sesquilinear form over (A_{L}, σ_{L}) obtained by extension of scalars: M_{L} = M ⊗_{K} L and s_{L}(x ⊗ a, y ⊗ b) = s(x, y) ⊗ ab for all x, y ∈ M and a, b ∈ L. Let ℛ be the category of reflexive right A-modules which are finite dimensional K-vector spaces.

7.1. Theorem Suppose that L/K is an extension of odd degree. Let (M, s) and (M′, s′) be two sesquilinear forms over (A, σ) defined on objects of ℛ. If (M_{L}, s_{L}) and (M′_{L}, s′_{L}) are isometric over (A_{L}, σ_{L}), then (M, s) and (M′, s′) are isometric over (A, σ).

In order to prove this result we will use techniques of hermitian categories. We denote by ℳ_{ℛ} ( (ℳ_{ℛ})_{L} ) the category of double arrows between objects of ℛ (respectively objects of ℛ tensorised with L over K). There is an obvious notion of scalar extension from the category ℳ_{ℛ} to the category (ℳ_{ℛ})_{L}: if Q = (V, W, f, g) is an object of ℳ_{ℛ}, set Q_{L} = (V_{L}, W_{L}, f_{L}, g_{L}), where V_{L} = V ⊗_{K} L, W_{L} = W ⊗_{K} L and the morphisms f_{L} and g_{L} are defined by extending f, respectively g to L.

Proof of theorem 7.1. The proof uses the equivalence of categories proven in §4. Consider the objects q(M, s) and q(M′, s′) of the category ℳ_{ℛ} and the objects q(M_{L}, s_{L}) and q(M′_{L}, s′_{L}) of the category (ℳ_{ℛ})_{L}. By hypothesis (M_{L}, s_{L}) and (M′_{L}, s′_{L}) are isometric, so by theorem 4.1, the objects q(M_{L}, s_{L}) ≃ q(M, s)_{L} and q(M′_{L}, s′_{L}) ≃ q(M′, s′)_{L} of (ℳ_{ℛ})_{L} are isomorphic. ¿From this it follows that the objects q(M, s) and q(M′, s′) of the category ℳ_{ℛ} are isomorphic.

Denote by Q the object q(M, s) of ℳ_{ℛ} and by E the endomorphism ring of Q in the category ℳ_{ℛ}.According to theorem 5.1, the set of isometry classes of sesquilinear forms (M′′, s′′) over (A, σ) such that q(M′′, s′′) ≃ Q is in bijection with the set H( , E^{×}). This last set is in turn in bijection with the set of isometry classes of rank one unimodular hermitian forms over (E, ). Denote by μ and μ′ the rank one unimodular hermitian forms over (E, ) that correspond by the above bijections to (M, s), respectively (M′, s′). Since (M_{L}, s_{L}) ≃ (M′_{L}, s′_{L}), the forms μ and μ′ extend to isometric hermitian forms over (E_{L}, _{L}), where E_{L} = E ⊗_{K} L and _{L} = ⊗ id_{L}. Applying Springer’s theorem for unimodular hermitian forms over (E, ) (cf. [4], th. 2.1), we deduce that μ ≃ μ′, hence (M, s) ≃ (M′, s′).

§8. Sesquilinear forms defined on algebras over complete discrete valuation rings

In this section we prove results similar to those of sections 6 and 7 for sesquilinear forms defined on algebras of finite rank over complete discrete valuation rings.

Let R be a complete discrete valuation ring and A be an R-algebra of finite rank with an involution denoted by σ. Suppose that there exists an element a in the center of A satisfying the property a + σ(a) = 1. For example this condition is fulfilled if 2 is invertible in R.

Let ℛ be the category of reflexive right A-modules of finite type.

8.1. Theorem Let (M, s), (M_{1}, s_{1}) and (M_{2}, s_{2}) be sesquilinear forms over (A, σ) defined on objects of ℛ such that

(M_{1}, s_{1}) ⊕ (M, s) ≃ (M_{2}, s_{2}) ⊕ (M, s).

It follows that (M_{1}, s_{1}) ≃ (M_{2}, s_{2}).

Proof Denote by ℳ_{ℛ} the category of double arrows between objects of ℛ, cf. §3. Due to the equivalence of categories between the categories 𝒮_{ℛ}(A, σ) and ℋ(ℳ_{ℛ}) given by theorem 4.1, it is enough to prove that Witt’s cancellation theorem holds in the category ℋ(ℳ_{ℛ}). This follows from theorem 6.2, observing that the category ℳ_{ℛ} satisfies conditions (i), (ii), (iii) (see e.g. [2], proposition 2).

Next we prove a result analogous to Springer’s theorem in the following setting: Let K be a non-dyadic local field, 𝒪_{K} be its ring of integers and A be an 𝒪_{K}- algebra of finite rank with an 𝒪_{K}-linear involution σ. Let L be a finite extension of K, 𝒪_{L} be its ring of integers and let A_{L} = A ⊗_{𝒪_{K}} 𝒪_{L} with the 𝒪_{L}-linear involution σ_{L} = σ ⊗ id_{𝒪_{L}}. If (M, s) is a sesquilinear form over (A, σ), then we denote by (M_{L}, s_{L}) the sesquilinear form over (A_{L}, σ_{L}) obtained by extension of scalars (in particular M_{L} = M ⊗_{𝒪_{K}} 𝒪_{L}).

Let ℛ be the category of reflexive right A-modules which are finite dimensional K-vector spaces.

8.2. Theorem Suppose that the extension L/K is of odd degree. Let (M, s) and (M′, s′) be two sesquilinear forms over (A, σ) with M, M′ ∈ ℛ. If (M_{L}, s_{L}) and (M′_{L}, s′_{L}) are isometric over (A_{L}, σ_{L}), then (M, s) and (M′, s′) are isometric over (A, σ).

Denote by ℳ_{ℛ} ( (ℳ_{ℛ})_{L} ) the category of double arrows between objects of ℛ (respectively objects of ℛ extended to 𝒪_{L}). There is an obvious notion of scalar extension from the category ℳ_{ℛ} to the category (ℳ_{ℛ})_{L}: for an object Q = (M, N, f, g) of ℳ_{ℛ} set Q_{L} = (M_{L}, N_{L}, f_{L}, g_{L}), where M_{L} = M ⊗_{𝒪_{K}} 𝒪_{L}, N_{L} = N ⊗_{𝒪_{K}} 𝒪_{L} and f_{L}, g_{L} are defined by extending f, respectively g. The proof of theorem 8.2 is analogous to the one of theorem 7.1, applying [6], p. 4 instead of [4], th. 2.1.

§9. Weak Hasse principle for sesquilinear forms

The classical Hasse-Minkowski theorem states that if two quadratic forms defined over a global field k of characteristic different from 2 become isometric over all the completions of k, then they are already isometric over k. The aim of this section is to investigate this result in the case of sesquilinear forms defined on a finite-dimensional skew field with involution. The case of bilinear forms has been treated by Waterhouse (cf. [14]), and is based on a classification result of Riehm (cf. [12]).

Let K be a field of characteristic different from 2, D be a finite-dimensional skew field with center K and σ be an involution on D. Denote by k the fixed field of σ in K. Then either k = K (when σ is said to be of the first kind) or K is a quadratic extension of k and the restriction of σ to K is the non-trivial automorphism of K over k (in which case σ is said to be of the second kind or a unitary involution).

Suppose that k is a global field. For every prime spot p of k, let k_{p} be the completion of k at p, K_{p} = K ⊗_{k} k_{p} and D_{p} = D ⊗_{k} k_{p}. Then D_{p} is an algebra with center K_{p} and we consider on it the extended involution σ_{p} = σ ⊗id_{k_{p}}. Then from any sesquilinear form (V, s) over (D, σ) we obtain by extension of scalars a sesquilinear form (V_{p}, s_{p}) over (D_{p}, σ_{p}), where V_{p} = V ⊗_{k} k_{p} is a free right D_{p}-module of rank dim_{D}(V).

We say that the weak Hasse principle holds for sesquilinear forms over (D, σ) if any two sesquilinear forms (V, s) and (V′, s′) over (D, σ) that become isometric over all the completions of k (i.e. (V_{p}, s_{p}) ≃ (V′_{p}, s′_{p}) over (D_{p}, σ_{p}) for every prime spot p of k) are already isometric over (D, σ).

In the sequel we will be interested in determining when the weak Hasse principle holds and we will adopt the point of view of hermitian categories. Let ℛ be a category of finite dimensional right D-vector spaces such that if V is an object of ℛ then V^{∗} is an object of ℛ as well. We denote by ℳ_{ℛ} ( (ℳ_{ℛ})_{p} ) the category of double arrows between objects of ℛ (respectively objects of ℛ extended to k_{p}). Let us fix a sesquilinear form s_{0} ∈ 𝒮_{ℛ}(D, σ). Consider a sesquilinear form s ∈ 𝒮_{ℛ}(D, σ) that becomes isometric to s_{0} over all the completions of k. Then for every prime spot p of k we have (s_{0})_{p} ≃ s_{p} and by theorem 4.1 it follows that q((s_{0})_{p}) ≃ q(s_{p}) for all p. Since q commutes with base change, we obtain q(s_{0})_{p} ≃ q(s)_{p}. It follows that q(s_{0}) ≃ q(s).

We denote by E the endomorphism ring of q(s_{0}) in ℳ_{ℛ}. The unimodular hermitian form η_{s_{0}} = (e_{V_{0}}, id_{V_{0}^{∗}}) defined on q(s_{0}) induces an involution on E. For every p let E_{p} = E ⊗_{k} k_{p}, on which we consider the involution _{p} = ⊗ id_{k_{p}}. By theorem 5.1 it is clear that if the localisation map

Φ : H( , E^{×}) → ∏_{p} H( _{p}, E_{p}^{×})

is injective, then the weak Hasse principle holds for sesquilinear forms in 𝒮_{ℛ}(D, σ). Hence in the sequel we will study the injectivity of the map Φ.

The involution induces an involution on E/rad(E), still denoted by . Since H( , E^{×}) ≃ H( , (E/rad(E))^{×}) (cf. [3], lemma 2.6), we can suppose that rad(E) = 0. The ring E being artinian, it is semi-simple. We can thus write E as a product of simple factors:

E ≃ ℳ_{n_{1}}(D_{1}) × ... × ℳ_{n_{r}}(D_{r}),

where for all 1 ≤ i ≤ r, D_{i} is a finite-dimensional skew field and n_{i} ≥ 1. Without loss of generality we can suppose that the first s factors are fixed by the involution and that the other factors are interchanged two by two. For every 1 ≤ i ≤ s we denote by σ_{i} the restriction of to ℳ_{n_{i}}(D_{i}). We then obtain the isomorphism

H( , E^{×}) ≃ ∏_{i=1}^{s} H(σ_{i}, ℳ_{n_{i}}(D_{i})^{×}).

Hence the map Φ is injective if and only if for every 1 ≤ i ≤ s the localisation map

Φ_{i} : H(σ_{i}, ℳ_{n_{i}}(D_{i})^{×}) → ∏_{p} H((σ_{i})_{p}, ℳ_{n_{i}}((D_{i})_{p})^{×}),

defined in the obvious way, is injective ((D_{i})_{p} = D_{i} ⊗_{k} k_{p} and (σ_{i})_{p} = σ ⊗ id_{k_{p}}). We thus obtain:

9.1. Theorem If for all 1 ≤ i ≤ s, either the involution σ_{i} is unitary, or D_{i} is a quaternion algebra and σ_{i} is a symplectic involution, then the map Φ is injective and hence the weak Hasse principle holds for sesquilinear forms over (D, σ).

Proof. It suffices to prove that for all 1 ≤ i ≤ s, the map Φ_{i} is injective. Let 1 ≤ i ≤ s and suppose that σ_{i} is unitary. According to the Albert-Riehm-Scharlau theorem ([9], theorem 3.1, p. 31), there exists a unitary involution τ_{i} on D_{i} with the same restriction to the center of D_{i} as σ_{i}. Then by Morita theory ([8], Chap. 1, theorem 9.3.5), the unimodular hermitian forms of rank one over (ℳ_{n_{i}}(D_{i}), σ_{i}) (which are parametrised by the set H(σ_{i}, ℳ_{n_{i}}(D_{i})) ) are in bijection with the unimodular hermitian forms of rank n_{i} over (D_{i}, τ_{i}). An analogous statement holds for the unimodular hermitian forms of rank one over (ℳ_{n_{i}}((D_{i})_{p}), (σ_{i})_{p}). As the weak Hasse principle holds for unimodular hermitian forms over a skew field with a unitary involution (cf. [13], theorem 10.6.1), the map Φ_{i} is injective.

Consider now the case when D_{i} is a quaternion algebra and σ_{i} is a symplectic involution. Then we consider the canonical involution τ_{i} on D_{i} and we proceed as above. The weak Hasse principle holds for unimodular hermitian forms over quaternion algebras with the canonical involution. Indeed, let h and h′ be two unimodular hermitian forms over (D_{i}, τ_{i}). Then their trace forms q_{h} and q_{h′} are two quadratic forms defined on the center of D_{i}. Since they become isometric everywhere locally, they are isometric by the usual weak Hasse principle. But two unimodular hermitian forms over (D_{i}, τ_{i}) are isometric if and only if their trace forms are isometric. Hence it follows that h ≃ h′. We conclude that the weak Hasse principle holds for unimodular hermitian forms over quaternion algebras with the canonical involution and thus the map Φ_{i} is injective.

9.2. Corollary If σ is a unitary involution, then the weak Hasse principle holds for sesquilinear forms in 𝒮_{ℛ}(D, σ).

Proof. The statement immediately follows from theorem 9.1, since if σ is unitary, then all the involutions σ_{i} are unitary as well.

§10. Finiteness results

In this section we generalize the finiteness results of [3] to the setting of sesquilinear forms. For a ring A we denote by T (A) the ℤ-torsion subgroup of A. If R is a ring then we say that A is R-finite if A_{R} = A ⊗_{ℤ} R is a finitely generated R-module and T (A) is finite.

Let (A, σ) be a ring with involution, ℛ be the category of reflexive right Amodules of finite type and ℳ_{ℛ} be the category defined in §3. We recall from §4 that the functor

ℱ : 𝒮_{ℛ}(A, σ) → ℋ(ℳ_{ℛ})
(V, s) ↦ ((V, V^{∗}, s_{l}, s_{r}), (e_{V} , id_{V^{∗}}))

is an equivalence of categories. Fix a sesquilinear form (V, s) ∈ 𝒮_{ℛ}(A, σ) and denote by q(V, s) the corresponding object (V, V^{∗}, s_{l}, s_{r}) of ℳ_{ℛ} and by E the endomorphism ring of q(V, s) in ℳ_{ℛ}.

10.1. Theorem If there exists a non-zero integer m such that End_{A}(V) is ℤ[1/m]-finite, then (V, s) contains only finitely many isometry classes of orthogonal summands.

Proof. From the fact that End_{A}(V) is ℤ[1/m]-finite it follows that E is ℤ[1/m]- finite and hence ℚ-finite. Hence by [3], theorem 1.1, q(V, s) has only finitely many isometry classes of direct summands.

Let (V′, s′) be an orthogonal summand of (V, s). Then q(V′, s′) is a direct summand of q(V, s). Since there is an obvious surjection End(q(V, s)) → End(q(V′, s′)), we obtain that End(q(V′, s′)) is ℤ[1/m]-finite too. By the next proposition there exist only finitely many isometry classes of sesquilinear forms (V′′, s′′) ∈ 𝒮_{ℛ}(A, σ) such that q(V′′, s′′) ≃ q(V′, s′). Hence the assertion of the theorem follows from the next result:

10.2. Proposition Let N be an object of ℳ_{ℛ} and assume that there exists a non-zero integer m such that End(N) is ℤ[1/m]-finite. Then there exist only finitely many isometry classes of sesquilinear forms (V, s) ∈ 𝒮_{ℛ}(A, σ) such that q(V, s) ≃ N.

Proof. By theorem 4.1, every sesquilinear form (V, s) ∈ 𝒮_{ℛ}(A, σ) defines a unimodular hermitian form on q(V, s). But by [3], theorem 1.2 the number of isometry classes of unimodular hermitian forms on N is finite. Hence the assertion follows.

We say that two sesquilinear forms over (A, σ) are in the same genus if they become isometric over A ⊗_{ℤ} ℤ_{p} for all primes of ℚ.

10.3. Theorem If End_{A}(V) is ℚ-finite then the genus of (V, s) contains only a finite number of isometry classes of sesquilinear forms.

Proof. Since End_{A}(V) is ℚ-finite, E is ℚ-finite too. Then by [3], theorem 3.4 the genus of ℱ(V, s) contains only finitely many isometry classes of unimodular hermitian forms. The functor ℱ induces a bijection between the genus of (V, s) and the genus of ℱ(V, s), hence the assertion follows.

§11. Bilinear forms invariant by a group action

Let R be a commutative ring and G be a finite group. Denote by R[G] the group ring of G over R. A G–bilinear form over R is by definition a pair (M, b), where M is a right R[G]–module which is an R-module of finite type and b : M ×M → R is an R–bilinear form such that b(xg, yg) = b(x, y) for all g ∈ G and x, y ∈ M. We say that two G–bilinear forms (M, b) and (M′, b′) are isometric if there exists an isomorphism of R[G]–modules f : M ˜→M′ such that b′(f(x), f(y)) = b(x, y) for all x, y ∈ M. Two G-bilinear forms over R are said to be in the same genus if they become isometric over R ⊗_{ℤ} ℤ_{p} for all primes of ℚ.

The aim of this section is to apply the results of sections 6, 7 and 10 to G–bilinear forms.

We endow the group ring R[G] with the canonical involution σ, which is the R-linear involution such that σ(g) = g^{−1} for every g ∈ G. Analogously to [7], theorem 7.1 it is straightforward to prove that there is a correspondence between G–bilinear forms over R and sesquilinear forms over (R[G], σ). Indeed, if (M, b) is a G–bilinear form over R then the form

S : M × M → R[G], (x, y) ↦ ∑_{g∈G} b(xg, y)g

for all x, y ∈ M is clearly sesquilinear with respect to σ. Conversely, if (M, S) is a sesquilinear form over (R[G], σ) then M has an obvious structure of R-module and the form P ∘ S : M × M → R is bilinear, where P is the projection of R[G] onto the e_{G} component.

11.1. Theorem Suppose that R is a field of characteristic ≠ 2. Let (M_{1}, b_{1}), (M_{2}, b_{2}) and (M, b) be G–bilinear forms over R such that (M_{1}, b_{1}) ⊕ (M, b) ≃ (M_{2}, b_{2}) ⊕ (M, b). Then (M_{1}, b_{1}) ≃ (M_{2}, b_{2}).

Proof. This follows from theorem 6.1 and the above correspondence.

11.2. Theorem Suppose that R is a field of characteristic ≠ 2. If two G– bilinear forms become isometric over an extension of R of odd degree then they are already isometric over R.

Proof. The result follows from theorem 7.1 and the above correspondence.

11.3. Theorem Let (M, b) be a G–bilinear form over R. If there exists a nonzero integer m such that End_{R[G]}(M) is ℤ[1/m]-finite, then (M, b) contains only finitely many isometry classes of orthogonal summands.

Proof. This follows from theorem 10.1 and the above correspondence.

11.4. Theorem Let (M, b) be a G–bilinear form over R. If End_{R[G]}(M) is ℚ-finite, then the genus of (M, b) contains only finitely many isometry classes of G–bilinear forms.

Proof. This follows from theorem 10.3 and the above correspondence.

§12. Generalization to systems of sesquilinear forms

Let A be a ring with an involution σ and I be a set. A system of sesquilinear forms over (A, σ) is (V, (s_{i})_{i∈I}), where V is a reflexive right A-module of finite type and for all i ∈ I, (V, s_{i}) is a sesquilinear form over (A, σ). A morphism between two systems of sesquilinear forms (V, (s_{i})_{i∈I}) and (V′, (s′_{i})_{i∈I}) consists of an isomorphism of A-modules f : V ˜→V′ such that for every i ∈ I and every x, y ∈ V we have s′_{i}(f(x), f(y)) = s_{i}(x, y).

Most of the results of the preceding sections can be generalized to systems of sesquilinear forms (for related results on systems of quadratic forms, see [1]). It is necessary to modify the definition of the category ℳ_{ℛ}: its objects will be of the form (V, W, (f_{i}, g_{i})_{i∈I}) and the morphisms between two such objects will be the obvious ones. To a system (V, (s_{i})_{i∈I}) of sesquilinear forms over (A, σ), where V ∈ ℛ, it will correspond the object (V, V^{∗}, (((s_{i})_{l}, (s_{i})_{r}))_{i∈I}) of the category ℳ_{ℛ}.

Acknowledgements The second named author would like to thank Emmanuel Lequeu for many interesting and useful discussions.

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

Daniel Arnold Moldovan École Polytechnique Fédérale de Lausanne EPFL-FSB-MATHGEOM-CSAG Station 8, 1015 Lausanne Switzerland danielarnold.moldovan@epfl.ch