Duality for partial group actions
Autor(a) principal: | |
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Data de Publicação: | 2008 |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
Texto Completo: | https://repositorio-aberto.up.pt/handle/10216/25719 |
Resumo: | Given a finite group G acting as automorphisms on a ring A, the skew group ring A*G is an important tool for studying the structure of G-stable ideals of A. The ring A*G is G-graded, i.e. G coacts on A*G. The Cohen-Montgomery duality says that the smash product A*G#k[G]^* of A*Gwith the dual group ring k[G]^* is isomorphic to the full matrix ring M_n(A) over A, where n is the order of G. In this note we show how much of the Cohen-Montgomery duality carries over to partial group actions alpha in the sense of R.Exel. In particular we show that the smash product (A *_alpha G)#k[G]^* of the partial skew group ring A*_alpha G and k[G]^* is isomorphic to a direct product of the form K x eM_n(A)e where e is a certain idempotent of M_n(A) and K isa subalgebra of (A *_alpha G)#k[G]^*. Moreover A*_alpha G is shown to be isomorphic to a separable subalgebra of eM_n(A)e. We also look at duality for infinite partial group actions. |
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Duality for partial group actionsÁlgebra, MatemáticaAlgebra, MathematicsGiven a finite group G acting as automorphisms on a ring A, the skew group ring A*G is an important tool for studying the structure of G-stable ideals of A. The ring A*G is G-graded, i.e. G coacts on A*G. The Cohen-Montgomery duality says that the smash product A*G#k[G]^* of A*Gwith the dual group ring k[G]^* is isomorphic to the full matrix ring M_n(A) over A, where n is the order of G. In this note we show how much of the Cohen-Montgomery duality carries over to partial group actions alpha in the sense of R.Exel. In particular we show that the smash product (A *_alpha G)#k[G]^* of the partial skew group ring A*_alpha G and k[G]^* is isomorphic to a direct product of the form K x eM_n(A)e where e is a certain idempotent of M_n(A) and K isa subalgebra of (A *_alpha G)#k[G]^*. Moreover A*_alpha G is shown to be isomorphic to a separable subalgebra of eM_n(A)e. We also look at duality for infinite partial group actions.20082008-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfhttps://repositorio-aberto.up.pt/handle/10216/25719eng1306-6048Christian Lompinfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-11-29T14:05:30Zoai:repositorio-aberto.up.pt:10216/25719Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T23:54:32.963347Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse |
dc.title.none.fl_str_mv |
Duality for partial group actions |
title |
Duality for partial group actions |
spellingShingle |
Duality for partial group actions Christian Lomp Álgebra, Matemática Algebra, Mathematics |
title_short |
Duality for partial group actions |
title_full |
Duality for partial group actions |
title_fullStr |
Duality for partial group actions |
title_full_unstemmed |
Duality for partial group actions |
title_sort |
Duality for partial group actions |
author |
Christian Lomp |
author_facet |
Christian Lomp |
author_role |
author |
dc.contributor.author.fl_str_mv |
Christian Lomp |
dc.subject.por.fl_str_mv |
Álgebra, Matemática Algebra, Mathematics |
topic |
Álgebra, Matemática Algebra, Mathematics |
description |
Given a finite group G acting as automorphisms on a ring A, the skew group ring A*G is an important tool for studying the structure of G-stable ideals of A. The ring A*G is G-graded, i.e. G coacts on A*G. The Cohen-Montgomery duality says that the smash product A*G#k[G]^* of A*Gwith the dual group ring k[G]^* is isomorphic to the full matrix ring M_n(A) over A, where n is the order of G. In this note we show how much of the Cohen-Montgomery duality carries over to partial group actions alpha in the sense of R.Exel. In particular we show that the smash product (A *_alpha G)#k[G]^* of the partial skew group ring A*_alpha G and k[G]^* is isomorphic to a direct product of the form K x eM_n(A)e where e is a certain idempotent of M_n(A) and K isa subalgebra of (A *_alpha G)#k[G]^*. Moreover A*_alpha G is shown to be isomorphic to a separable subalgebra of eM_n(A)e. We also look at duality for infinite partial group actions. |
publishDate |
2008 |
dc.date.none.fl_str_mv |
2008 2008-01-01T00:00:00Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://repositorio-aberto.up.pt/handle/10216/25719 |
url |
https://repositorio-aberto.up.pt/handle/10216/25719 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
1306-6048 |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
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openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.source.none.fl_str_mv |
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Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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RCAAP |
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RCAAP |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) |
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Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação |
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