Group cohomology based on partial representations
Autor(a) principal: | |
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Data de Publicação: | 2020 |
Tipo de documento: | Dissertação |
Idioma: | eng |
Título da fonte: | Biblioteca Digital de Teses e Dissertações da USP |
Texto Completo: | https://www.teses.usp.br/teses/disponiveis/45/45131/tde-06102020-125952/ |
Resumo: | We consider the partial group cohomology $H_^n(G,M)$ of a group $G$ with values in $\\K_G$-module $M$, which is defined as the right derived functor of the functor of partial invariants. Showing that the functor of partial invariants is representable, we relate the partial group cohomology with the space of partial derivations and the partial augmentation ideal; next, we construct a projective resolution of the algebra $B$ as a $\\K_G$-module, where $B$ is a commutative subalgebra of $\\K_G$. This allows us to give another characterization of the partial group cohomology in terms of classes of functions that satisfy a certain identity of $n$-cocycles. We show the existence of a Grothendieck spectral sequence that relates the cohomology of the partial smash product with the partial group cohomology and the algebra cohomology. Given a unital partial action $\\alpha$ of $G$ on a algebra $\\mathcal$ we consider the $\\K_G$-module structure of $\\mathcal$ induced by $\\alpha$ and study the globalization problem for the partial cohomology with values in $\\mathcal$. The problem is reduced to an extendibility property of cocycles. Moreover, if $\\mathcal$ is a product of indecomposable blocks, we show that any cocycle is globalizable, and globalizations of cohomologous cocycles are also cohomologous, whence we have that $H_^n(G,M)$ is isomorphic to the usual cohomology group $H^n(G, \\mathcal(\\mathcal))$, where $\\mathcal$ is the algebra under the enveloping action of $\\alpha$ and $\\mathcal(\\mathcal)$ is the multiplier algebra of $\\mathcal$. |
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Group cohomology based on partial representationsCohomologia de grupo baseada em representações parciaisAção parcialCohomologiaCohomologyGlobalizaçãoGlobalizationPartial actionPartial smash productProduto parcial smashSequência espectralSpectral sequenceWe consider the partial group cohomology $H_^n(G,M)$ of a group $G$ with values in $\\K_G$-module $M$, which is defined as the right derived functor of the functor of partial invariants. Showing that the functor of partial invariants is representable, we relate the partial group cohomology with the space of partial derivations and the partial augmentation ideal; next, we construct a projective resolution of the algebra $B$ as a $\\K_G$-module, where $B$ is a commutative subalgebra of $\\K_G$. This allows us to give another characterization of the partial group cohomology in terms of classes of functions that satisfy a certain identity of $n$-cocycles. We show the existence of a Grothendieck spectral sequence that relates the cohomology of the partial smash product with the partial group cohomology and the algebra cohomology. Given a unital partial action $\\alpha$ of $G$ on a algebra $\\mathcal$ we consider the $\\K_G$-module structure of $\\mathcal$ induced by $\\alpha$ and study the globalization problem for the partial cohomology with values in $\\mathcal$. The problem is reduced to an extendibility property of cocycles. Moreover, if $\\mathcal$ is a product of indecomposable blocks, we show that any cocycle is globalizable, and globalizations of cohomologous cocycles are also cohomologous, whence we have that $H_^n(G,M)$ is isomorphic to the usual cohomology group $H^n(G, \\mathcal(\\mathcal))$, where $\\mathcal$ is the algebra under the enveloping action of $\\alpha$ and $\\mathcal(\\mathcal)$ is the multiplier algebra of $\\mathcal$.Consideraremos a cohomologia parcial $H_^n(G, M)$ de um grupo $G$ com valores num $K_G$-módulo $M$, introduzida em \\cite, que é definida como o functor derivado à direita do functor de invariantes parciais. Mostrando que o functor de invariantes parciais é representável, poderemos relacionar a cohomologia parcial de grupo com o espaço de derivações parciais e o ideal de aumento parcial; depois, construiremos uma resolução projetiva da álgebra $B$ como $K_G$-modulo, onde $B$ é una subálgebra de $K_G$. Isto permitirá dar uma outra caracterização da cohomologia parcial de grupo em termos de classes de funções que satisfazem uma certa identidade de $n$-cociclos. Mostramos a existência de uma sequência espectral de Grothendieck que relaciona a cohomologia do produto smash parcial com a cohomologia parcial do grupo e a cohomologia da álgebra. Dada uma ação parcial unital $\\alpha$ de $G$ em uma álgebra $\\mathcal$, consideramos a estrutura de $K_G$-módulo de $\\mathcal$ induzida pela ação $\\alpha$ e estudamos o problema de globalização para a cohomologia parcial em $\\mathcal$. O problema é reduzido a uma propriedade de extensibilidade de cociclos. Além disso, se $\\mathcal$ é um produto de blocos, mostramos que qualquer cociclo é globalizável e que as globalizações de cociclos cohomólogos também são cohomólogas, de onde temos que $H_^n(G,M)$ é isomórfico ao grupo de cohomologia usual $H^n(G,\\mathcal(\\mathcal))$, onde $\\mathcal$ é a álgebra sob a ação envolvente de $\\alpha$ e $\\mathcal(\\mathcal)$ é a álgebra de multiplicadores de $\\mathcal$.Biblioteca Digitais de Teses e Dissertações da USPDokuchaev, MikhailoUsuga, Emmanuel Jerez2020-08-21info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisapplication/pdfhttps://www.teses.usp.br/teses/disponiveis/45/45131/tde-06102020-125952/reponame:Biblioteca Digital de Teses e Dissertações da USPinstname:Universidade de São Paulo (USP)instacron:USPLiberar o conteúdo para acesso público.info:eu-repo/semantics/openAccesseng2021-01-21T01:13:02Zoai:teses.usp.br:tde-06102020-125952Biblioteca Digital de Teses e Dissertaçõeshttp://www.teses.usp.br/PUBhttp://www.teses.usp.br/cgi-bin/mtd2br.plvirginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.bropendoar:27212021-01-21T01:13:02Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)false |
dc.title.none.fl_str_mv |
Group cohomology based on partial representations Cohomologia de grupo baseada em representações parciais |
title |
Group cohomology based on partial representations |
spellingShingle |
Group cohomology based on partial representations Usuga, Emmanuel Jerez Ação parcial Cohomologia Cohomology Globalização Globalization Partial action Partial smash product Produto parcial smash Sequência espectral Spectral sequence |
title_short |
Group cohomology based on partial representations |
title_full |
Group cohomology based on partial representations |
title_fullStr |
Group cohomology based on partial representations |
title_full_unstemmed |
Group cohomology based on partial representations |
title_sort |
Group cohomology based on partial representations |
author |
Usuga, Emmanuel Jerez |
author_facet |
Usuga, Emmanuel Jerez |
author_role |
author |
dc.contributor.none.fl_str_mv |
Dokuchaev, Mikhailo |
dc.contributor.author.fl_str_mv |
Usuga, Emmanuel Jerez |
dc.subject.por.fl_str_mv |
Ação parcial Cohomologia Cohomology Globalização Globalization Partial action Partial smash product Produto parcial smash Sequência espectral Spectral sequence |
topic |
Ação parcial Cohomologia Cohomology Globalização Globalization Partial action Partial smash product Produto parcial smash Sequência espectral Spectral sequence |
description |
We consider the partial group cohomology $H_^n(G,M)$ of a group $G$ with values in $\\K_G$-module $M$, which is defined as the right derived functor of the functor of partial invariants. Showing that the functor of partial invariants is representable, we relate the partial group cohomology with the space of partial derivations and the partial augmentation ideal; next, we construct a projective resolution of the algebra $B$ as a $\\K_G$-module, where $B$ is a commutative subalgebra of $\\K_G$. This allows us to give another characterization of the partial group cohomology in terms of classes of functions that satisfy a certain identity of $n$-cocycles. We show the existence of a Grothendieck spectral sequence that relates the cohomology of the partial smash product with the partial group cohomology and the algebra cohomology. Given a unital partial action $\\alpha$ of $G$ on a algebra $\\mathcal$ we consider the $\\K_G$-module structure of $\\mathcal$ induced by $\\alpha$ and study the globalization problem for the partial cohomology with values in $\\mathcal$. The problem is reduced to an extendibility property of cocycles. Moreover, if $\\mathcal$ is a product of indecomposable blocks, we show that any cocycle is globalizable, and globalizations of cohomologous cocycles are also cohomologous, whence we have that $H_^n(G,M)$ is isomorphic to the usual cohomology group $H^n(G, \\mathcal(\\mathcal))$, where $\\mathcal$ is the algebra under the enveloping action of $\\alpha$ and $\\mathcal(\\mathcal)$ is the multiplier algebra of $\\mathcal$. |
publishDate |
2020 |
dc.date.none.fl_str_mv |
2020-08-21 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/masterThesis |
format |
masterThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://www.teses.usp.br/teses/disponiveis/45/45131/tde-06102020-125952/ |
url |
https://www.teses.usp.br/teses/disponiveis/45/45131/tde-06102020-125952/ |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
|
dc.rights.driver.fl_str_mv |
Liberar o conteúdo para acesso público. info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Liberar o conteúdo para acesso público. |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.coverage.none.fl_str_mv |
|
dc.publisher.none.fl_str_mv |
Biblioteca Digitais de Teses e Dissertações da USP |
publisher.none.fl_str_mv |
Biblioteca Digitais de Teses e Dissertações da USP |
dc.source.none.fl_str_mv |
reponame:Biblioteca Digital de Teses e Dissertações da USP instname:Universidade de São Paulo (USP) instacron:USP |
instname_str |
Universidade de São Paulo (USP) |
instacron_str |
USP |
institution |
USP |
reponame_str |
Biblioteca Digital de Teses e Dissertações da USP |
collection |
Biblioteca Digital de Teses e Dissertações da USP |
repository.name.fl_str_mv |
Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP) |
repository.mail.fl_str_mv |
virginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.br |
_version_ |
1809090457770131456 |