000177326 001__ 177326 000177326 005__ 20181205220131.0 000177326 0247_ $$2doi$$a10.5075/epfl-thesis-5371 000177326 02470 $$2urn$$aurn:nbn:ch:bel-epfl-thesis5371-7 000177326 02471 $$2nebis$$a7273398 000177326 037__ $$aTHESIS 000177326 041__ $$aeng 000177326 088__ $$a5371 000177326 245__ $$aHermitian Forms over Algebras with Involution and Hermitian Categories 000177326 269__ $$a2012 000177326 260__ $$aLausanne$$bEPFL$$c2012 000177326 300__ $$a160 000177326 336__ $$aTheses 000177326 520__ $$aThis 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. 000177326 6531_ $$aunimodular [epsilon]-hermitian forms 000177326 6531_ $$asesquilinear forms 000177326 6531_ $$aWitt groups 000177326 6531_ $$arestriction map 000177326 6531_ $$aalgebras with involution 000177326 6531_ $$aMorita theory 000177326 6531_ $$adescent properties 000177326 6531_ $$ahermitian categories 000177326 6531_ $$alinear hermitian categories 000177326 6531_ $$aHasse-Minkowski principle 000177326 6531_ $$aformes [epsilon]-hermitiennes unimodulaires 000177326 6531_ $$aformes sesquilinéaires 000177326 6531_ $$agroupes de Witt 000177326 6531_ $$al'application de restriction 000177326 6531_ $$aalgèbres à involution 000177326 6531_ $$athéorie de Morita 000177326 6531_ $$apropriétés de descente 000177326 6531_ $$acatégories hermitiennes 000177326 6531_ $$acatégories hermitiennes linéaires 000177326 6531_ $$aprincipe de Hasse-Minkowski 000177326 700__ $$0244949$$aMoldovan, Daniel Arnold$$g184998 000177326 720_2 $$0244699$$aBayer Fluckiger, Eva$$edir.$$g138858 000177326 8564_ $$s955052$$uhttps://infoscience.epfl.ch/record/177326/files/EPFL_TH5371.pdf$$yTexte intégral / Full text$$zTexte intégral / Full text 000177326 909C0 $$0252344$$pCSAG$$xU10800 000177326 909CO $$ooai:infoscience.tind.io:177326$$pDOI$$pthesis$$pthesis-bn2018$$qDOI2 000177326 918__ $$aSB$$cMATHGEOM$$dEDMA 000177326 919__ $$aCSAG 000177326 920__ $$b2012 000177326 970__ $$a5371/THESES 000177326 973__ $$aEPFL$$sPUBLISHED 000177326 980__ $$aTHESIS