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