Hermitian categories, extension of scalars and systems of sesquilinear forms

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

December 29, 2012

Abstract In this paper we define a notion of Witt group for sesquilinear forms in hermitian categories, which in turn provides a notion of Witt group for sesquilinear forms over rings with involution. We also study the extension of scalars for K-linear hermitian categories, where K is a field of characteristic ≠ 2. We finally extend several results concerning sesquilinear forms to the setting of systems of such forms.

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

Keywords sesquilinear forms, hermitian forms, systems of sesquilinear forms, hermitian categories, K-linear categories, scalar extension, Witt group.

Introduction

There exists a classical notion of Witt group for hermitian forms over rings with involution (see for instance [6] and [8]). However, there is no analogous notion for sesquilinear forms. In this paper we define, based on an equivalence of categories proven in [5], a notion of Witt group for sesquilinear forms in hermitian categories. In particular we obtain a notion of Witt group for sesquilinear forms over a ring with involution.

We then study the extension of scalars for K-linear hermitian categories, where K is a field of characteristic ≠ 2, and we prove its injectivity in the case of an extension of odd degree and a hermitian category in which all idempotents split.

We also introduce the notion of system of sesquilinear forms over a ring with involution and we generalize several results proven in [5] to systems of sesquilinear forms.

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

§1. Sesquilinear and hermitian 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) over (A, σ) induces two homomorphisms of left Amodules 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. We observe that s_{r} = s_{l}^{∗}e_{V}.

Let ϵ = ±1. A sesquilinear form (V, s) over (A, σ) is called ϵ-hermitian if V is a projective A-module and σ(s(x, y)) = ϵs(y, x) for all x, y ∈ V, i.e. s_{l} = ϵs_{r}. A 1-hermitian form is also called a hermitian form. An ϵ-hermitian form (V, s) is called unimodular if s_{l} (or equivalently s_{r}) is bijective. There exists a classical notion of Witt group for unimodular ϵ-hermitian forms over (A, σ) (see e.g. [6]). Denote by Gr^{ϵ}(A, σ) the Grothendieck group of isometry classes of unimodular ϵ-hermitian forms over (A, σ), with respect to the orthogonal sum. A unimodular ϵ-hermitian form over (A, σ) is called hyperbolic if it is isometric to a form ℍ(V), where V is a finitely generated projective right A-module and

ℍ(V) : V ⊗ V^{∗} → V^{∗} ⊕ V^{∗∗}
x + y ↦ y + ϵe_{V} (x), ∀x ∈ V, y ∈ V^{∗}.

The quotient of Gr^{ϵ}(A, σ) by the subgroup generated by the unimodular ϵhermitian forms is called the Witt group of unimodular ϵ-hermitian forms over (A, σ) and is denoted by W^{ϵ}(A, σ).

Let us denote by 𝒮(A, σ) ( ℋ^{ϵ}(A, σ) ) the category of sesquilinear (respectively unimodular ϵ-hermitian) forms over (A, σ). The morphisms of these categories are isometries. For simplicity let ℋ(A, σ) = ℋ^{1}(A, σ).

§2. Hermitian categories

The aim of this section is to recall some basic notions about hermitian categories as presented in [8] (see also [6], [7]).

§2.1. Preliminaries

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. An additive category with a duality functor is called a hermitian category. A sesquilinear form in the category 𝒞 is a pair (C, s), where C is an object of 𝒞 and s : C → C^{∗}. A sesquilinear form (C, s) is called unimodular if s is an isomorphism. Let ϵ = ±1. An ϵ-hermitian form in the category 𝒞 is a sesquilinear form (C, s) such that s = ϵs^{∗}E_{C}. A 1-hermitian form is also called a hermitian form. Orthogonal sums of forms are defined in the obvious way. Let (C, s) and (C′, s′) be two sesquilinear forms in 𝒞. We say that these forms are isometric if there exists an isomorphism f : C ˜→C′ in the category 𝒞 such that s = f^{∗}s′f.

Denote by ℋ^{ϵ}(𝒞) the category of unimodular ϵ-hermitian forms in the category 𝒞. The morphisms are isometries. For simplicity let ℋ(𝒞) = ℋ^{1}(𝒞).

The hyperbolic unimodular ϵ-hermitian forms in 𝒞 are the forms isometric to ℍ_{Q}, Q ∈ 𝒞, given by

ℍ_{Q} = ϵE_{Q} ⊕ id_{Q^{∗}} : Q ⊕ Q^{∗} → Q^{∗∗} ⊕ Q^{∗} ≃ (Q ⊕ Q^{∗})^{∗}.

The quotient of the Grothendieck group of isometry classes of unimodular ϵhermitian forms in 𝒞 (with respect to the orthogonal sum) by the subgroup generated by the hyperbolic forms is called the Witt group of unimodular ϵhermitian forms in 𝒞 and is denoted by W^{ϵ}(𝒞). For simplicity set W (𝒞) = W^{1}(𝒞).

We observe that if we take 𝒞 to be the category of reflexive right A-modules of finite type, then the notion of sesquilinear form coincides with the one defined in the preceding section. Analogously, if we take 𝒞 to be the category of projective right A-modules of finite type, then the notion of hermitian form coincides with the one defined in the preceding section.

Let (ℳ, ∗) and (ℳ′, ∗) be two hermitian categories. A duality preserving functor from (ℳ, ∗) to (ℳ′, ∗) is an additive functor F : ℳ → ℳ′ together with a natural isomorphism i = (i_{M})_{M∈ℳ} : F^{∗} → ^{∗}F. This means that for any M ∈ ℳ there exists a natural isomorphism i_{M} : F (M^{∗}) ˜→F (M)^{∗} such that for all N ∈ ℳ and f ∈ Hom_{ℳ}(M, N) the following diagram commutes:

\xymatrix{F (N^{∗}) \ar[r]^{F (f^{∗})} \ar[d]^{i_{N}} & F (M^{∗}) \ar[d]^{i_{M}} \\ F (N)^{∗} \ar[r]^{F (f)^{∗}} & F (M)^{∗}}

For λ = ±1, a duality preserving functor F is called λ-hermitian if i_{M^{∗}}F (e_{M}) = λi_{M}^{∗}e_{F (M)} for all M ∈ ℳ. Let ϵ = ±1. We recall from [6], pp. 80-81 that a λ-hermitian functor F : ℳ → ℳ′ induces a functor

Herm^{ϵ}(F) : ℋ^{ϵ}(ℳ) → ℋ^{ϵλ}(ℳ′)
(M, h) ↦ (F (M), i_{M}F (h)),

which preserves orthogonal sums and hyperbolicity. Therefore it induces a homomorphism between the corresponding Witt groups:

\overline{Herm}^{ϵ}(F) : W^{ϵ}(ℳ) → W^{ϵλ}(ℳ′).

If F is fully faithful, then Herm^{ϵ}(F) is also fully faithful. Moreover, if F is an equivalence of categories, then Herm^{ϵ}(F) is also an equivalence of categories and the induced group homomorphism \overline{Herm}^{ϵ}(F) is bijective.

§2.2. Transfer into the endomorphism ring

The aim of this subsection is to introduce the method of transfer into the endomorphism ring, which allows us to pass from the abstract setting of hermitian categories to that of a ring with involution, which is more concrete. This method will be extensively applied in section 4.

Let ℳ be a hermitian category and M be an object of ℳ, on which we suppose that there exists a unimodular ϵ_{0}-hermitian form h_{0} for a certain ϵ_{0} = ±1. Denote by E the endomorphism ring of M. According to [7], lemma 1.2, the form (M, h) induces on E an involution σ, defined by σ(f) = h_{0}^{−1}f^{∗}h_{0} for all f ∈ E.

We say that an idempotent e ∈ E splits if there exist an object M′ ∈ ℳ and morphisms i : M′ → M, j : M → M′ such that ji = id_{M′} and ij = e.

Denote by ℳ|_{M} the full subcategory of ℳ which has as objects all the objects of ℳ isomorphic to a direct summand of a finite direct sum of copies of M and by 𝒫(E) the category of finitely generated projective right E-modules. We consider the following functor:

ℱ = Hom(M, −) : ℳ|_{M} → 𝒫(E)
N ↦ Hom(M, N), ∀N ∈ ℳ|_{M}

f ↦ ℱ(f), ∀f ∈ Hom(N, N′), ∀N, N′ ∈ ℳ|_{M},

where for all g ∈ Hom(M, N), ℱ(f)(g) = fg. In [7], proposition 2.4 it has been proven that this functor is fully faithful and duality preserving with respect to the natural isomorphism i = (i_{N})_{N∈ℳ|_{M}} : ℱ^{∗} → ^{∗}ℱ given by i_{N}(f) = ℱ(h_{0}^{−1}f^{∗}e_{N}) for every N ∈ ℳ|_{M} and f ∈ Hom(M, N^{∗}). In addition, if all the idempotents of ℳ|_{N} split, then ℱ is an equivalence of categories. By computation we easily see that ℱ is ϵ_{0}-hermitian.

§2.3. Ring extension and genus in hermitian categories

In this subection we introduce the notions of extension of rings and genus in hermitian categories.

Let 𝒞 be an additive category and R be a commutative ring with unity. By extension of scalars from ℤ to R we obtain a new category, denoted by 𝒞 ⊗_{ℤ} R and called the extension of the category 𝒞 to the ring R. Its objects are the same as those of 𝒞 and for two such objects M and N set

Hom_{𝒞⊗_{ℤ}R}(M, N) = Hom_{𝒞}(M, N) ⊗_{ℤ} R.

It is straightforward to check that in this way we obtain an additive category. If, in addition, there is a duality functor ∗ in the category 𝒞, then 𝒞 ⊗_{ℤ} R becomes a hermitian category by setting (f ⊗a)^{∗} = f^{∗}⊗a for all M, N ∈ 𝒞, f ∈ Hom_{𝒞}(M, N) and a ∈ R.

For each prime spot p of ℚ we denote by 𝒞_{p} the category 𝒞 ⊗_{ℤ} ℤ_{p}.

Let 𝒞 be an additive category. Two sesquilinear forms s : M → M^{∗} and t : N → N^{∗} in the category 𝒞 are said to be in the same genus if for all primes p of ℚ there exist F_{p} ∈ Hom_{𝒞}(M, N)_{p} and G_{p} ∈ Hom_{𝒞}(N, M)_{p} such that F_{p}G_{p} and G_{p}F_{p} are equal to the identity and in addition F_{p}^{∗}(t ⊗ 1)F_{p} = s ⊗ 1. This is equivalent to saying that the extended sesquilinear forms (M, s ⊗ 1) and (N, t ⊗ 1) become isometric in the extended category 𝒞_{p} for every p.

§2.4. K-linear hermitian categories and scalar extension

The aim of this section is to give an introduction to the theory of K-linear hermitian categories. Their consideration is motivated by the idea of defining a notion of scalar extension in hermitian categories.

2.4.1. Definition Let K be a field. A K-linear category is an additive category ℳ such that for every M, N ∈ ℳ, the set Hom_{ℳ}(M, N) has a structure of finite-dimensional K-vector space such that the composition of morphisms is K-bilinear. An additive functor F : ℳ → 𝒩 between two K-linear categories is called K-linear if for any M, N ∈ ℳ, the induced map

F_{M,N} : Hom_{ℳ}(M, N) → Hom_{𝒩}(F (M), F (N))

is K-linear.

A K-linear hermitian category is a K-linear category together with a K-linear duality functor.

Let (ℳ, ∗) be a K-linear hermitian category and consider a finite field extension L of K. We define the extension of the category ℳ to L as being the category ℳ_{L} with the same objects as ℳ and with the morphisms given by

Hom_{ℳ_{L}}(M, N) = Hom_{ℳ}(M, N) ⊗_{K} L

for all M, N ∈ ℳ. It is clear that the category ℳ_{L} is L-linear.

If in ℳ there is a K-linear duality functor ∗ then we can define a duality functor in ℳ_{L} in the following way: on objects it coincides with the duality functor of ℳ and for morphisms set (f ⊗ a)^{∗} = f^{∗} ⊗ a for all f ∈ Hom_{ℳ}(M, N), M, N ∈ ℳ and a ∈ L and then extend by additivity.

The scalar extension functor from ℳ to ℳ_{L} is defined by

ℛ_{L/K} : ℳ → ℳ_{L}
M → M, ∀M ∈ ℳ
f ↦ f ⊗ 1, ∀f ∈ Hom(M, N).

It is straightforward to check that the functor ℛ_{L/K} is 1-hermitian. As we have seen in §2.1, it induces for every ϵ = ±1 a group homomorphism

\overline{Herm}^{ϵ}(ℛ_{L/K}) : W^{ϵ}(ℳ) → W^{ϵ}(ℳ_{L}),

called restriction map.

§3. Sesquilinear forms over rings with involution and hermitian categories

In this section we prove that the category of sesquilinear forms over a ring with involution is equivalent with the category of unimodular hermitian forms in a suitably constructed hermitian category. We also define a notion of Witt group for sesquilinear forms in hermitian categories. In particular we obtain a notion of Witt group for sesquilinear forms over rings with involution, which generalizes the analogous notion for unimodular hermitian forms.

§3.1. An equivalence of categories

Let ℳ be an additive category. On the model of [5], §3, we construct the category of double arrows of ℳ, denoted by ℳ^{(2)}. Its objects are of the form (M, N, f, g), where M, N ∈ ℳ and f, g ∈ Hom(M, N). A morphism from (M, N, f, g) to (M′, N′, f′, g′) is a pair (ϕ, ψ), where ϕ ∈ Hom(M, M′) and ψ ∈ Hom(N, N′) satisfying the conditions ψf = f′ϕ and ψg = g′ϕ. Then ℳ^{(2)} is obviously an additive category. If in addition there is a duality functor ∗ in ℳ, then ℳ^{(2)} becomes a hermitian category by defining the dual of (M, N, f, g) as being (N^{∗}, M^{∗}, g^{∗}, f^{∗}) and setting E_{(M,N,f,g)} = (e_{M}, e_{N}).

We define a functor F : 𝒮(ℳ) → ℋ(ℳ^{(2)}) as follows. Let (M, s) be a sesquilinear form in ℳ. Then (M, M^{∗}, s, s^{∗}e_{M}) is an object of ℳ^{(2)} and it is easy to check that (e_{M}, id_{M^{∗}}) defines a unimodular hermitian form on it. Set F (M, s) = ((M, M^{∗}, s, s^{∗}e_{M}), (e_{M}, id_{M^{∗}})). For an isometry ϕ : (M, s) → (M′, s′) of sesquilinear forms in ℳ set F (ϕ) = (ϕ, ϕ^{∗−1}).

We also define a functor G in the opposite direction: for every object (M, N, f, g) of ℳ^{(2)} and any unimodular hermitian form (ξ_{1}, ξ_{2}) on it set G((M, N, f, g), (ξ_{1}, ξ_{2})) = (M, ξ_{2}f). For a morphism (λ_{1}, λ_{2}) between two unimodular hermitian forms in ℳ^{(2)} define G(λ_{1}, λ_{2}) = λ_{1}.

3.1.1. Theorem The functors F and G realize an equivalence between the categories 𝒮(ℳ) and ℋ(ℳ^{(2)}).

Proof. The proof is completely analogous to the one of [5], theorem 4.1.

§3.2. Hyberbolic sesquilinear forms

Using the functor G we can define a notion of hyperbolicity for sesquilinear forms in ℳ, which will in turn lead to a notion of Witt group.

3.2.1. Definition A sesquilinear form in ℳ is called hyperbolic if it is isometric to a form G(P, s), where (P, s) ∈ ℋ(ℳ^{(2)}) is hyperbolic.

Let (A, σ) be a ring with involution. Taking ℳ to be the category of reflexive right A-modules of finite type we obtain a notion of hyperbolicity for sesquilinear forms over (A, σ).

We have the following explicit characterization of hyperbolic sesquilinear forms over (A, σ):

3.2.2 Proposition A sesquilinear form (V, s) over (A, σ) is hyperbolic if and only if there exist two reflexive right A-modules of finite type M and N and two A-linear homomorphisms f, g : M → N such that (V, s) is isometric to the form (M ⊕ N^{∗}, H̃_{(M,N,f,g)}), given by

H̃_{(M,N,f,g)} : (M ⊕ N^{∗}) × (M ⊕ N^{∗}) → A

(x_{1} + y_{1}, x_{2} + y_{2}) ↦ y_{1}(g(x_{2})) + σ(y_{2}(f(x_{1}))), ∀x_{1}, x_{2} ∈ M, ∀y_{1}, y_{2} ∈ N^{∗}.

Proof. For all Q = (M, N, f, g) ∈ ℳ^{(2)} the hyperbolic unimodular hermitian form ℍ_{Q} in ℳ^{(2)} is defined by

ℍ_{Q} = E_{Q} ⊕ id_{Q^{∗}} : Q ⊕ Q^{∗} → Q^{∗∗} ⊕ Q^{∗} ≃ (Q ⊕ Q^{∗})^{∗},

hence G(ℍ_{Q}) is given by

G(ℍ_{Q}) : M ⊕ N^{∗} → M^{∗} ⊕ N^{∗∗}
x + y ↦ g^{∗}(y) + e_{N}(f(x)), ∀x ∈ M, ∀y ∈ N^{∗}.

It is easy to identify this form with H̃_{Q}. □

Particularizing the last proposition, we obtain:

3.2.3. Proposition

a) The hyperbolic hermitian forms over (A, σ) are given, up to isometry, by the forms (M ⊕N^{∗}, H̃_{(M,N,f,f)}), where M and N are two reflexive right A-modules of finite type and f : M → N is an A-linear homomorphism.

b) The hyperbolic unimodular sesquilinear forms over (A, σ) are given, up to isometry, by (M ⊕ N^{∗}, H̃_{(M,N,f,g)}), where M and N are two reflexive right A-modules of finite type and f, g : M → N are two A-linear isomorphisms.

c) The hyperbolic unimodular hermitian forms over (A, σ) are given, up to isometry, by (M ⊕ M^{∗}, ℍ_{M}), where M is a reflexive right A-module of finite type and

ℍ_{M} : (M ⊕ M^{∗}) × (M ⊕ M^{∗}) → A
(x_{1} + y_{1}, x_{2} + y_{2}) ↦ y_{1}(x_{2}) + σ(y_{2}(x_{1})), ∀x_{1}, y_{1} ∈ M, ∀y_{1}, y_{2} ∈ M^{∗}.

Proof. It is straightforward to check that the form (M ⊕ N^{∗}, H̃_{(M,N,f,g)}) is hermitian if and only if f = g and that it is unimodular if and only if f and g are isomorphisms. The point c) follows from a) and b) and from the fact that the forms (M ⊕N^{∗}, H̃_{(M,N,f,f)}) and (M ⊕M^{∗}, ℍ_{M}) are isometric via the isomorphism of A-modules id_{M} ⊕ f^{∗} : M ⊕ N^{∗} ˜→M ⊕ M^{∗}. □

3.2.4. Definition Let Gr_{S}(ℳ) be the Grothendieck group of isometry classes of sesquilinear forms in ℳ, with respect to the orthogonal sum. It is easy to check that the isometry classes of hyperbolic sesquilinear forms form a subgroup of Gr_{S}(ℳ), which we denote by ℍ_{S}(ℳ). The Witt group of sesquilinear forms in the category ℳ is defined to be the quotient

W_{S}(ℳ) = Gr_{S}(ℳ)/ℍ_{S}(ℳ).

Taking ℳ to be the category of reflexive right A-modules of finite type we obtain a notion of Witt group for sesquilinear forms over (A, σ). We observe that in the case of unimodular hermitian forms our definition coincides with the well-known one.

§3.3. Finiteness results concerning the genus of sesquilinear forms

In this subsection we prove a finiteness result concerning the genus of sesquilinear forms, based on the methods of [5], §9. For a ring A we denote by T (A) the ℤtorsion subgroup of A. If R is a ring containing ℤ, 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.

3.3.1. Theorem Let A be a ring and σ be an involution on A. Let (V, s) be a sesquilinear form over (A, σ) and assume that End_{A}(V) is ℚ-finite. Then the genus of (V, s) contains only a finite number of isometry classes of sesquilinear forms.

Proof. The functor F defined in §3.1 induces a bijection between the genus of (V, s) and the genus of F (V, s). Denote by E the endomorphism ring of (V, V^{∗}, s_{l}, s_{r}) in ℳ. Since End_{A}(V) is ℚ-finite and E is a subring of End_{A}(V) × End_{A}(V^{∗}) ≃ End_{A}(V) × End_{A}(V), E is ℚ-finite too. ¿From [3], theorem 3.4 it follows that the genus of F (V, s) contains only a finite number of isometry classes of unimodular hermitian forms, which implies that the genus of (V, s) contains only a finite number of isometry classes of sesquilinear forms.

§4. The restriction map for K-linear hermitian categories in odd degree extensions

It is well-known that if K is a field of characteristic ≠ 2 and L/K is an extension of odd degree, then the restriction map r_{L/K} : W (K) → W (L) is injective. The aim of this section is to prove an analogous result for K-linear hermitian categories.

Let K be a field of characteristic ≠ 2 and ℳ be a K-linear hermitian category. Let L be a finite extension of K, ℳ_{L} be the extension of the category ℳ to L and ℛ_{L/K} : ℳ → ℳ_{L} be the scalar extension functor, as defined in §2.5.

4.1. Theorem Suppose that all the idempotents of ℳ split and that the extension L/K is of odd degree. Then for all ϵ = ±1 the map

\overline{Herm}^{ϵ}(ℛ_{L/K}) : W^{ϵ}(ℳ) → W^{ϵ}(ℳ_{L})

is injective.

This result will follow as an immediate corollary from the following one.

4.2. Theorem Let M be an object of ℳ that admits a unimodular hermitian or skew-hermitian form. Suppose that all the idempotents of ℳ|_{M} split and that the extension L/K is of odd degree. Then for all ϵ = ±1, the map

\overline{Herm}^{ϵ}(ℛ_{L/K}^{M}) : W^{ϵ}(ℳ|_{M}) → W^{ϵ}(ℳ_{L}|_{M})

is injective.

Proof of theorem 4.2 Denote by h_{0} a unimodular ϵ_{0}-hermitian form on M. Since the category ℳ is K-linear, the endomorphism ring E of M in ℳ has a structure of finite-dimensional K-algebra. According to [7], lemma 1.2, h_{0} induces on E a K-linear involution, denoted by σ. Clearly the endomorphism ring E′ of M in ℳ_{L} equals E ⊗_{K} L and is a finite-dimensional L-algebra. We also observe that (M, h_{0L}) is a unimodular ϵ_{0}-hermitian form in ℳ_{L} and that it induces by [7], lemma 1.2 the involution σ ⊗ id_{L} on E′. We have an obvious 1-hermitian functor of extension of scalars

ℛ_{L/K}^{M} : ℳ|_{M} → ℳ_{L}|_{M}.

We recall from §2.2 that there are two fully faithful ϵ_{0}-hermitian functors

ℱ : ℳ|_{M} → 𝒫(E),
ℱ′ : ℳ_{L}|_{M} → 𝒫(E ⊗_{K} L).

We denote by R_{L/K} the obvious functor of scalar extension

R_{L/K} : 𝒫(E) → 𝒫(E ⊗_{K} L).

It is straightforward to prove that the following diagram commutes:

\xymatrix{ℳ_{L}|_{M} \ar[r]^{ℱ′} & 𝒫(E ⊗_{K} L) \\ ℳ|_{M} \ar[r]^{ℱ} \ar[u]^{ℛ_{L/K}^{M}} & 𝒫(E) \ar[u]_{R_{L/K}}}

For ϵ = ±1 the above diagram induces a commutative diagram on the level of unimodular hermitian forms:

\xymatrix{ℋ^{ϵ}(ℳ_{L}|_{M}) \ar[r]^{Herm^{ϵ}(ℱ′)} & ℋ^{ϵϵ_{0}}(E ⊗_{K} L, σ ⊗ id_{L}) \\ ℋ^{ϵ}(ℳ|_{M}) \ar[r]^{Herm^{ϵ}(ℱ)} \ar[u]^{Herm^{ϵ}(ℛ_{L/K}^{M})} & ℋ^{ϵϵ_{0}}(E, σ) \ar[u]_{Herm^{ϵϵ_{0}}(R_{L/K})}}

and one on the level of Witt groups:

\xymatrix{W^{ϵ}(ℳ_{L}|_{M}) \ar[r]^{\overline{Herm}^{ϵ}(ℱ′)} & W^{ϵϵ_{0}}(E ⊗_{K} L, σ ⊗ id_{L}) \\ W^{ϵ}(ℳ|_{M}) \ar[r]^{\overline{Herm}^{ϵ}(ℱ)} \ar[u]^{\overline{Herm}_{ϵ}(ℛ_{L/K}^{M})} & W^{ϵϵ_{0}}(E, σ) \ar[u]_{\overline{Herm}_{ϵϵ_{0}}(R_{L/K})}}

Let (N, g) and (N′, g′) ∈ ℋ^{ϵ}(ℳ|_{M}) be such that

\overline{Herm}^{ϵ}(ℛ_{L/K}^{M})(N, g) = \overline{Herm}^{ϵ}(ℛ_{L/K}^{M})(N′, g′)

in W^{ϵ}((ℳ_{L})|_{M}). It follows that

\overline{Herm}^{ϵ}(ℱ′)(\overline{Herm}^{ϵ}(ℛ_{L/K}^{M})(N, g)) = \overline{Herm}^{ϵ}(ℱ′)(\overline{Herm}^{ϵ}(ℛ_{L/K}^{M})(N′, g′))

in W^{ϵϵ_{0}}(E ⊗_{K} L, σ ⊗ id_{L}). As the last diagram above commutes, we obtain

\overline{Herm}^{ϵϵ_{0}}(R_{L/K})(\overline{Herm}^{ϵ}(ℱ)(N, g)) = \overline{Herm}^{ϵϵ_{0}}(R_{L/K})(\overline{Herm}^{ϵ}(ℱ)(N′, g′)).

By [4], proposition 1.2, the map \overline{Herm}^{ϵϵ_{0}}(R_{L/K}) is injective, so we deduce that \overline{Herm}^{ϵ}(ℱ)(N, g) = \overline{Herm}^{ϵ}(ℱ)(N′, g′) in W^{ϵϵ_{0}}(E, σ). Since in the category ℳ|_{M} all idempotents split, the functor ℱ is an equivalence of categories and hence the group homomorphism \overline{Herm}^{ϵ}(ℱ) : W^{ϵ}(ℳ|_{M}) → W^{ϵϵ_{0}}(E, σ) is bijective (see §2.1). It follows that (N, g) = (N′, g′) in W^{ϵ}(ℳ|_{M}). We conclude that the map \overline{Herm}^{ϵ}(ℛ_{L/K}^{M}) is injective.

Proof of theorem 4.1 Let (N, g) ∈ ℋ^{ϵ}(ℳ) be such that \overline{Herm}^{ϵ}(ℛ_{L/K})(N, g) is hyperbolic. This means that (N, g ⊗ 1) is a hyperbolic unimodular ϵ-hermitian form in the category ℳ_{L}, so in the category ℳ_{L}|_{N} too. Hence \overline{Herm}^{ϵ}(ℛ_{L/K}^{N})(N, g) = 0 and from to the injectivity of the map \overline{Herm}^{ϵ}(ℛ_{L/K}^{N}) we deduce that (N, g) is hyperbolic in ℳ|_{N}, so in ℳ too. In conclusion the map \overline{Herm}^{ϵ}(ℛ_{L/K}) is injective.

§5. Systems of sesquilinear forms

In this section we generalize results proven in [5] to systems of sesquilinear forms. We first construct an equivalence of categories as in [5], §4.

§5.1. Preliminaries

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 x, y ∈ V we have s′_{i}(f(x), f(y)) = s_{i}(x, y). Let us denote by 𝒮^{(I)}(A, σ) the category of systems of sesquilinear forms over (A, σ).

Denote by J the disjoint union of two copies of I. We define the category of J-arrows between reflexive A-modules as being the category ℳ^{(J)} constructed in the following way: its objects are of the form (V, W, (f_{i}, g_{i})_{i∈I}), where V and W are two reflexive A-modules of finite type and for all i ∈ I, f_{i}, g_{i} : V → W are homomorphisms of A-modules. A morphism from (V, W, (f_{i}, g_{i})_{i∈I}) to (V′, W′, (f′_{i}, g′_{i})_{i∈I}) is a pair (ϕ, ψ), where ϕ : V ˜→V′ and ψ : W ˜→W′ are isomorphisms of A-modules such that for all i ∈ I we have ψf_{i} = f′_{i}ϕ and ψg_{i} = g′_{i}ϕ. By defining direct sums in the obvious way we see that ℳ^{(J)} is an additive category. Let (W^{∗}, V^{∗}, (g_{i}^{∗}, f_{i}^{∗})_{i∈I}) be the dual of (V, W, (f_{i}, g_{i})_{i∈I}) and set E_{(V,W,(f_{i},g_{i})_{i∈I})} = (e_{V} , e_{W}). This defines a duality on the category ℳ^{(J)}. We observe that if the set I has one element, then ℳ^{(J)} coincides with the category constructed in [5], §3.

We define a functor Ψ : 𝒮^{(I)}(A, σ) → ℳ^{(J)} in the following way: Let (V, (s_{i})_{i∈I}) be an object of 𝒮^{(I)}(A, σ) and for all i ∈ I let s_{il} : V → V^{∗} and s_{ir} : V → V^{∗} be the left, respectively the right adjoint of (V, s_{i}) (cf. §1). Then (V, V^{∗}, (s_{il}, s_{ir})_{i∈I}) is an object of ℳ^{(J)} and it is easy to check that (e_{V} , id_{V^{∗}}) defines a unimodular hermitian form on it. Let Ψ(V, (s_{i})_{i∈I}) = ((V, V^{∗}, (s_{il}, s_{ir})_{i∈I}), (e_{V} , id_{V^{∗}})). For a morphism φ : (V, (s_{i})_{i∈I}) → (V′, (s′_{i})_{i∈I}) of systems of sesquilinear forms over (A, σ) set Ψ(φ) = (φ, φ^{∗−1}).

5.1.1. Theorem The functor Ψ is an equivalence of categories between 𝒮^{(I)}(A, σ) and ℋ(ℳ^{(J)}).

Proof. The proof is analogous to the one of [5], theorem 4.1. The functor Φ : ℋ(ℳ^{(J)}) → 𝒮(A, σ) which realizes, together with Ψ, the desired equivalence of categories is defined in the following way: Let ((V, W, (f_{i}, g_{i})_{i∈I}), (φ, ψ)) be an object of ℋ(ℳ^{(J)}). For every i ∈ I define a sesquilinear form s_{i} : V × V → A by s_{i}(x, y) = (ψg_{i})(x)(y) for all x, y ∈ V and set Φ((V, W, (f_{i}, g_{i})_{i∈I}), (φ, ψ)) = (V, (s_{i})_{i∈I}). For a morphism λ = (λ_{1}, λ_{2}) between two objects of ℋ(ℳ^{(J)}) set Φ(λ) = λ_{1}.

If (V, (s_{i})_{i∈I}) is a system of sesquilinear forms over (A, σ), then we denote by q(V, (s_{i})_{i∈I}) the corresponding object (V, V^{∗}, (s_{il}, s_{ir})_{i∈I}) of the category ℳ^{(J)}. We next describe, following [5], §5, the set of isometry classes of systems of sesquilinear forms over (A, σ) corresponding by theorem 5.1.1 to a given object of the category ℳ^{(J)}.

Let us fix a system of sesquilinear forms (V_{0}, (s_{0i})_{i∈I}) over (A, σ) and consider the unimodular hermitian form η_{V_{0}} = (e_{V_{0}}, id_{V_{0}^{∗}}) on Q_{0} = q(V_{0}, (s_{0i})_{i∈I}). Let E be the endomorphism ring of the object Q_{0} in ℳ^{(J)}. The form η_{V_{0}} induces an involution on E, defined by f̃ = η_{V_{0}}^{−1}f^{∗}η_{V_{0}} for all f ∈ E, where f^{∗} denotes the dual of f in ℳ^{(J)}. As in [5], §5 we denote by H( , E^{×}) the set of equivalence classes for the equivalence relation defined on {f ∈ E| f̃ = f} by: f ≡ f′ if there exists a g ∈ E^{×} such that g̃fg = f′. Analogously to [5], theorem 5.1 we obtain:

5.1.2. Theorem The set of isometry classes of systems of sesquilinear forms (V, (s_{i})_{i∈I}) over (A, σ) such that q(V, (s_{i})_{i∈I}) ≃ Q_{0} is in bijection with H( , E^{×}).

Proof. The proof is analogous to the one of [5], theorem 5.1. For every system of sesquilinear forms (V, (s_{i})_{i∈I}) over (A, σ) such that q(V, (s_{i})_{i∈I}) ≃ Q_{0} the unimodular hermitian form (e_{V} , id_{V^{∗}}) on q(V, (s_{i})_{i∈I}) induces a unimodular hermitian form η_{V} on Q_{0}. The desired bijection is given by:

{[(V, (s_{i})_{i∈I})] | (V, (s_{i})_{i∈I}) ∈ 𝒮^{(I)}(A, σ), q(V, (s_{i})_{i∈I}) ≃ Q_{0}} → H( , E^{×})
[(V, (s_{i})_{i∈I})] ↦ η_{V_{0}}^{−1}η_{V} .

§5.2. Witt’s cancellation theorem

Let K be a field of characteristic ≠ 2, A be a finite-dimensional K-algebra and σ be an involution on A. Analogously to [5], theorem 6.1, a cancellation theorem holds for systems of sesquilinear forms over (A, σ):

5.2.1. Theorem Let (V, (s_{i})_{i∈I}), (V′, (s′_{i})_{i∈I}) and (V′′, (s′′_{i})_{i∈I}) be systems of sesquilinear forms over (A, σ) such that

(V′, (s′_{i})_{i∈I}) ⊕ (V, (s_{i})_{i∈I}) ≃ (V′′, (s′′_{i})_{i∈I}) ⊕ (V, (s_{i})_{i∈I}).

Then we have (V′, (s′_{i})_{i∈I}) ≃ (V′′, (s′′_{i})_{i∈I}).

Due to the equivalence between the categories 𝒮^{(I)}(A, σ) and ℋ(ℳ^{(J)}) given by theorem 5.1.1, it is enough to prove that Witt’s cancellation theorem holds in the category ℋ(ℳ^{(J)}). This can be proven as in [5], proposition 6.2.

§5.3. Springer’s theorem

In this subsection we prove an analogue of Springer’s theorem for systems of sesquilinear forms defined over finite-dimensional algebras with involution.

Let K be a field of characteristic ≠ 2, A be a finite-dimensional K-algebra and σ be a K-linear involution on A. 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 (V, (s_{i})_{i∈I}) is a system of sesquilinear forms over (A, σ), then we denote by (V, (s_{i})_{i∈I})_{L} = (V_{L}, ((s_{i})_{L})_{i∈I}) the system of sesquilinear forms over (A_{L}, σ_{L}) obtained by extension of scalars.

5.3.1. Theorem Suppose that L/K is an extension of odd degree. Let (V, (s_{i})_{i∈I}) and (V′, (s′_{i})_{i∈I}) be two systems of sesquilinear forms over (A, σ). If (V, (s_{i})_{i∈I})_{L} and (V′, (s′_{i})_{i∈I})_{L} are isometric over (A_{L}, σ_{L}), then (V, (s_{i})_{i∈I}) and (V′, (s′_{i})_{i∈I}) are isometric over (A, σ).

Proof. The proof is analogous to the one of [5], theorem 7.1 and uses the equivalence of categories given by theorem 5.1.1.

§5.4. Weak Hasse principle

The well-known weak Hasse principle states that if two quadratic forms defined over a global field k of characteristic ≠ 2 become isometric over all the completions of k, then they are already isometric over k. The aim of this section is to generalize this result to the case of systems of sesquilinear forms defined over a skew field with involution. Throughout this section we suppose that I is finite.

For results concerning the weak Hasse principle for systems of hermitian or quadratic forms over fields see [1], respectively [2]. A weak Hasse principle for sesquilinear forms defined over a skew field with involution has been proven in [5].

Let K be a field of characteristic ≠ 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 consider on it the 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). Hence we obtain a notion of extension of scalars for systems of sesquilinear forms over (D, σ) by setting (V, (s_{i})_{i∈I})_{p} = (V_{p}, ((s_{i})_{p})_{i∈I}).

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

For the problem of determining when the weak Hasse principle holds we will use the ideas developed in [5]. We denote by ℳ^{(J)} ( (ℳ_{p})^{(J)} ) the category of J-arrows between reflexive D– (resp. D_{p}–) modules. Let us fix a system of sesquilinear forms s_{0} over (D, σ). Consider a system of sesquilinear forms s over (D, σ) that becomes isometric to s_{0} over all the completions of k. Denote by q(s) the object of ℳ^{(J)} corresponding to s by the equivalence of categories given in theorem 5.1.1. For every prime spot p of k we have (s_{0})_{p} ≃ s_{p}, hence q((s_{0})_{p}) ≃ q(s_{p}). Since q commutes with base change, we obtain q(s_{0})_{p} ≃ q(s)_{p} and since I is finite, it follows that q(s_{0}) ≃ q(s).

Denote by E the endomorphism ring of q(s_{0}) in ℳ. 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 5.1.2 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 systems of sesquilinear forms over (D, σ). As in [5], §8, we prove the following:

5.4.1. Theorem If σ is a unitary involution, then the map Φ is injective and hence the weak Hasse principle holds for systems of sesquilinear forms over (D, σ).

§5.5. Finiteness results

In this section we generalize theorem 3.3.1 and the results proven in [5], §9 to systems of sesquilinear forms. Following §2.3 we say that two systems of sesquilinear forms over (A, σ) are in the same genus if they become isometric over all the extensions of A to ℤ_{p}, where p is a prime.

Fix a system of sesquilinear forms (V, (s_{i})_{i∈I}) over (A, σ) and denote by q(V, (s_{i})_{i∈I}) the corresponding object (V, V^{∗}, (s_{il}, s_{ir})_{i∈I}) of ℳ^{(J)} and by E its endomorphism ring in ℳ^{(J)}. It is straightforward to prove the following results using the methods of [5], §9 and the equivalence of categories F : 𝒮^{(I)}(A, σ) → ℋ(ℳ^{(J)}) given by theorem 5.1.1:

5.5.1. Theorem If there exists a non-zero integer m such that End_{A}(V) is ℤ[1/m]-finite, then there exist only finitely many isometry classes of systems of sesquilinear forms on V.

5.5.2. Theorem Let N be an object of ℳ^{(J)} and assume that there exists a non-zero integer m such that End_{ℳ^{(J)}}(N) is ℤ[1/m]-finite. Then there exist only finitely many isometry classes of systems of sesquilinear forms (V, (s_{i})_{i∈I}) over (A, σ) such that q(V, (s_{i})_{i∈I}) ≃ N.

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

5.5.4. Theorem If End_{A}(V) is ℚ-finite, then the genus of (V, (s_{i})_{i∈I}) contains only a finite number of isometry classes of systems of sesquilinear forms.

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

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