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Concept# Bilinear form

Summary

In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately:

- B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v)
- B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v)

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Let R be a semilocal Dedekind domain. Under certain assumptions, we show that two (not necessarily unimodular) hermitian forms over an R-algebra with involution, which are rationally isomorphic and have isomorphic semisimple coradicals, are in fact isomorphic. The same result is also obtained for quadratic forms equipped with an action of a finite group. The results have cohomological restatements that resemble the Grothendieck-Serre conjecture, except the group schemes involved are not reductive. We show that these group schemes are closely related to group schemes arising in Bruhat-Tits theory. (C) 2017 Elsevier Inc. All rights reserved.

Eva Bayer Fluckiger, Daniel Arnold Moldovan

We prove that the category of systems of sesquilinear forms over a given hermitian category is equivalent to the category of unimodular 1-hermitian forms over another hermitian category. The sesquilinear forms are not required to be unimodular or defined on a reflexive object (i.e., the standard map from the object to its double dual is not assumed to be bijective), and the forms in the system can be defined with respect to different hermitian structures on the given category. This extends an earlier result of the first and third authors. We use the equivalence to define a Witt group of sesquilinear forms over a hermitian category and to generalize results such as Witt's cancellation theorem, Springer's theorem, the weak Hasse principle, and finiteness of genus to systems of sesquilinear forms over hermitian categories.

This thesis is concerned with the algebraic theory of hermitian forms. It is organized in two parts. The first, consisting of the first two chapters, deals with some descent properties of unimodular hermitian forms over central simple algebras with involution. The second, which consists of the last two chapters, generalizes several classical properties of unimodular hermitian forms over rings with involution to the setting of sesquilinear forms in hermitian categories. The original results established in this thesis are joint work with Professor Eva Bayer-Fluckiger. The first chapter contains an introduction to the algebraic theory of unimodular ε-hermitian forms over fields with involution. One knows that if L/K is an extension of odd degree (where char(K) ≠ 2) then the restriction map rL/K : W(K) →W(L) is injective. In addition, if the extension is purely inseparable then the map rL/K is bijective. In the second chapter we first introduce the basic notions and techniques of the theory of unimodular ε-hermitian forms over algebras with involution, in particular the technique of Morita equivalence. Let L/K be a finite field extension, τ an involution on L and A a finite-dimensional K-algebra endowed with an involution α such that αœK = τœK. E. Bayer-Fluckiger and H.W. Lenstra proved that if L/K is of odd degree and αœK = idK then the restriction map rL/Kε : Wε(A, α) → Wε(A ⊗K L, α ⊗ τ) is injective for any ε = ±1. This holds also if αœK ≠ idK. We prove that if, in addition, L/K is purely inseparable and A is a central simple K-algebra, then the above map is actually bijective. The proof proceeds via induction on the degree of the algebra and uses in an essential way an exact sequence of Witt groups due to R. Parimala, R. Sridharan and V. Suresh, later extended by N. Gernier-Boley and M.G. Mahmoudi. The third chapter contains a survey of the theory of hermitian and quadratic forms in hermitian categories. In particular, we cover the transfer between two hermitian categories, the reduction by an ideal, the transfer into the endomorphism ring of an object, as well as the Krull-Schmidt-Azumaya theorem and some of its applications. In the fourth chapter we prove, adapting the ideas developed by E. Bayer-Fluckiger and L. Fainsilber, that the category of sesquilinear forms in a hermitian category ℳ is equivalent to the category of unimodular hermitian forms in the category of double arrows of ℳ. In order to obtain this equivalence of categories we associate to a sesquilinear form the double arrow consisting of its two adjoints, equipped with a canonical unimodular hermitian form. This equivalence of categories allows us to define a notion of Witt group for sesquilinear forms in hermitian categories. This generalizes the classical notion of a Witt group of unimodular hermitian forms over rings with involution. Using the above equivalence of categories we deduce analogues of the Witt cancellation theorem and Springer's theorem for sesquilinear forms over certain algebras with involution. We also extend some finiteness results due to E. Bayer-Fluckiger, C. Kearton and S.M. J. Wilson. In addition, we study the weak Hasse-Minkowski principle for sesquilinear forms over skew fields with involution over global fields. We prove that this principle holds for systems of sesquilinear forms over a skew field over a global field and endowed with a unitary involution. Two systems of sesquilinear forms are hence isometric if and only if they are isometric over all the completions of the base field. This result has already been known for unimodular hermitian and skew-hermitian forms over rings with involution, under the same hypothesis. Finally, we study the behaviour of the Witt group of a linear hermitian category under extension of scalars. Let K be a field of characteristic different from 2, L a finite extension of K and ℳ a K-linear hermitian category. We define the extension of ℳ to L as being the category with the same objects as ℳ and with morphisms given by the morphisms of ℳ extended to L. We obtain an L-linear hermitian category, denoted by ℳL. The canonical functor of scalar extension ℛL/K : ℳ → ℳL induces for any ε = ±1 a group homomorphism Wε(ℳ) →Wε(ℳL). We prove that if all the idempotents of the category ℳ split and the extension L/K is of odd degree then this homomorphism is injective. This result has already been known in the case when ℳ is the category of finite-dimensional K-vector spaces.