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Master's Dissertation
DOI
10.11606/D.55.2017.tde-29032017-085640
Document
Author
Full name
Wellington Marques de Souza
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2017
Supervisor
Committee
Pérez, Victor Hugo Jorge (President)
Brusamarello, Rosali
Levcovitz, Daniel
Talpo, Humberto Luiz
Title in Portuguese
Sequências espectrais e aplicações para módulos
Keywords in Portuguese
Álgebra
Complexos
Homologia
Sequências Espectrais
Abstract in Portuguese
As sequências espectrais foram criadas por Jean Leray num campo de concentração durante a Segunda Guerra Mundial motivado por problemas inerentes à Topologia Algébrica. Num primeiro momento, surge como uma ferramenta para auxiliar no cálculo da cohomologia de um feixe. Porém, Jean-Louis Koszul apresenta uma formulação puramente algébrica para tais sequencias, que consiste basicamente no cálculo da homologia de um complexo total associado a um complexo duplo. Concentraremos nosso trabalho nas definições e resultados que nos permitem demonstrar os seguintes resultados conhecidos da Álgebra usando sequências espectrais: o Lema dos Cinco, o Lema da Serpente, Balanceamento para o Funtor Tor, Mudança de Base para o Funtor Tor e o Teorema dos Coeficientes Universais. Apresentamos, ao final do trabalho, uma generalização que nos permite entender melhor os funtores derivados à esquerda: as Sequências Espectrais de Grothendieck.
Title in English
Spectral sequences and applications to modules
Keywords in English
Algebra
Complex
Homology
Spectral sequences
Abstract in English
Spectral sequences were created by Jean Leray in a concentration camp during World War II motivated by problems of Algebraic Topology. At first, it appears as a tool to assist in calculating the cohomology of a sheaf. However, Jean-Louis Koszul presents a purely algebraic formulation for these sequences, which basically consists in calculating a total of homology complex associated with a double complex. We will focus our work on the definitions and results that allow us to demonstrate known results of algebra using spectral sequences: The Five Lemma, The Snake Lemma, Balancing of functor Tor, Base Change for Tor and Universal Coefficient Theorem. We present, at the end of this work, a generalization that allows us to better understand the left derivative functors: the Spectral Sequence of Grothendieck.
 
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Publishing Date
2017-03-29
 
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