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Master's Dissertation
DOI
https://doi.org/10.11606/D.55.2020.tde-10062020-103904
Document
Author
Full name
Alexandre Carissimi
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2020
Supervisor
Committee
Pérez, Victor Hugo Jorge (President)
Borges Filho, Herivelto Martins
Dantas, Alex Carrazedo
Miranda Neto, Cleto Brasileiro
Title in Portuguese
Álgebra homológica e cohomologia de grupos
Keywords in Portuguese
Extensões de grupo
Funtores Ext e Tor
Módulos projetivos e injetivos
Abstract in Portuguese
Neste trabalho abordamos conceitos básicos de teoria de categorias e aplicamos tais ideias à categoria de módulos sobre um anel. Também desenvolvemos as ferramentas necessárias para se estudar álgebra homológica, como complexos de cadeia, resoluções projetivas e injetivas, para então tratar dos funtores Ext e Tor. Em seguida, utilizamos tais construções para definir a cohomologia de um grupo G com coeficientes em um G-módulo M, calculamos alguns grupos de cohomologia nos níveis baixos e damos um procedimento padrão para se obter uma resolução projetiva do grupo abeliano dos números inteiros visto como G-módulo trivial. Finalmente, aplicamos estes conceitos para abordar o problema da extensão de grupos, dando uma caracterização das extensões de um grupo abeliano M por um grupo qualquer G usando a cohomologia de grupos.
Title in English
Homological algebra and group cohomology
Keywords in English
Ext and Tor functors
Group extensions
Projective and injective modules
Abstract in English
In this work we approach basic concepts of category theory and apply these ideas to the category of modules over a given ring. We also develop the needed tools to study homological algebra, e.g. chain complexes and projective and injective resolutions and then we treat the Ext and Tor functors. After that, we use such constructions to define the cohomology of a group G with coefficients on a G-module M, we calculate some low level cohomology groups and give a standard procedure to obtain a projective resolution of the abelian group of the integers viewed as a trivial G-module. Finally, we apply these concepts to approach the problem of the group extensions, giving a characterization of the extensions of an abelian group M by a group G using group cohomology.
 
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Publishing Date
2020-06-10
 
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