000177326 001__ 177326
000177326 005__ 20190118220121.0
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