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Master's Dissertation
DOI
10.11606/D.45.2009.tde-29072009-192529
Document
Author
Full name
Francisco Batista de Medeiros
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2009
Supervisor
Committee
Marcos, Eduardo do Nascimento (President)
Cortes, Wagner de Oliveira
Goldschmidt, Hector Alfredo Merklen
Title in Portuguese
Álgebras de Koszul e resoluções projetivas
Keywords in Portuguese
álgebra de extensões
álgebras de Koszul
bases de Gröbner
representações de álgebras.
resoluções lineares
resoluções projetivas
Abstract in Portuguese
Neste trabalho estudamos algumas características das álgebras de Koszul, como por exemplo, a maneira como elas se relacionam com suas respectivas álgebras de Yoneda. Descrevemos a álgebra de Yoneda de uma álgebra monomial e como aplicação construímos uma família de álgebras: as chamadas homologicamente auto-duais. Uma álgebra de Koszul pode ser definida a partir da existência de resoluções lineares dos módulos simples. Por isso faz-se necessário a dedicação de parte de nossa atenção ao estudo destas resoluções. Além disso, achamos interessante estudar métodos para a construção de resoluções projetivas de módulos sobre quocientes de álgebras de caminhos. Para tal construção usamos essencialmente a teoria de bases de Gröbner não comutativas. Finalmente, para o caso de módulos lineares sobre álgebras de Koszul, veremos que é possível modicar essa construção de modo que a resolução resultante seja linear.
Title in English
Koszul algebras and projetive resolutions
Keywords in English
algebra of extensions
Gröbner bases
Koszul algebras
linear resolutions
projetive resolutions
representation of algebras.
Abstract in English
In this work we study some features of Koszul algebras as, for example, the way that they are related with their Yoneda algebras. We describe the Yoneda algebra of a monomial algebra and as an application we construct a family of algebras: the so called homologically self-dual algebras. A Koszul algebra can be dened as an algebra for which there are linear resolutions of their simple modules. Because of this we dedicate part of our attention to the study of projective resolutions. The study of methods for the construction of projectives resolutions of modules over quotients of path algebras, has an of interest its own. For the study of projective resolutions we used the theory of noncommutative, Gröbner bases. Finally, for the case of linear modules over Koszul algebras, we will see that it is possible to modify the general construction described here, so that the resulting resolution is linear.
 
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Publishing Date
2009-10-13
 
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