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Master's Dissertation
DOI
10.11606/D.45.2016.tde-22062016-104927
Document
Author
Full name
Duilio Ferreira Santos
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2015
Supervisor
Committee
Mariano, Hugo Luiz (President)
Lopes, Vinicius Cifú
Miraglia Neto, Francisco
Title in Portuguese
Elementos da teoria algébrica das formas quadráticas e de seus anéis graduados
Keywords in Portuguese
Anel de Witt
Cohomologia galoisiana
Conjectura de Marshall
Corpos
Formas quadráticas
K-teoria de Milnor
Abstract in Portuguese
Neste trabalho procuramos realizar uma apresentação autocontida sobre os conceitos da teoria algébrica de formas quadráticas e sobre os anéis graduados que surgiram no desenvolvimento desta teoria. Iniciamos procurando esclarecer o sentido da equivalência entre as várias acepções do conceito de forma quadrática. Após a apresentação de ingredientes e resultados geométricos, fazemos um extrato da teoria dos anéis de Witt, conceito que originou a moderna teoria algébrica de formas quadráticas. Disponibilizamos os elementos fundamentais para a formulação das teorias de cohomologia, nos concentrado no desenvolvimento da teoria de cohomologia profinita e, sobretudo, galoisiana. Descrevemos os funtores K0, K1 e K2 da K-teoria clássica e também a K-teoria de Milnor, que é mais adequada para formular questões sobre formas quadráticas. Finalizamos o trabalho com a apresentação de alguns conceitos da Teoria dos Grupos Especiais, uma codificação em primeira-ordem da teoria algébrica das formas quadráticas e exemplificamos sua importância, fornecendo um extrato da prova realizada por Dickmann-Miraglia da conjectura de Marshall sobre assinaturas, que se baseia fortemente nesta teoria.
Title in English
Elements of the algebraic theory of quadratic forms and its graded rings
Keywords in English
Fields
Galois cohomology
Marshalls conjecture
Milnor K-theory
Quadratic forms
Witt rings
Abstract in English
In this work I try to provide a self-contained presentation on the concepts of algebraic theory of quadratic forms and on the graded rings that have emerged in the development of this theory. I started trying to clarify the meaning of "equivalence"between the various meanings of the concept of quadratic form. After the presentation of geometrical ingredients and results, we make an extract of the theory of Witt rings, a concept that originated the modern algebraic theory of quadratic forms. It is provided the key elements for the formulation of cohomology theories, focusing on the development of profinite cohomology theory and, especially, on galoisian cohomology. Are described the functors K0, K1 and K2 of classical K-theory and also the Milnor K-theory, which is more appropriate to formulate questions about quadratic forms. The dissertation is finished with the presentation of some concepts of the Theory of Special Groups, a first-order encoding of algebraic theory of quadratic forms, and with an example its importance by providing an extract of proof by Dickmann-Miraglia of the Marshalls conjecture on signatures, which relies heavily on this theory.
 
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disspadrao.pdf (1.69 Mbytes)
Publishing Date
2016-08-29
 
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