On a definition of multi-Koszul algebras

Autores
Herscovich Ramoneda, Estanislao Benito; Rey, Andrea
Año de publicación
2013
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this article we introduce the notion of multi-Koszul algebra for the case of a nonnegatively graded connected algebra with a finite number of generators of degree 1 and with a finite number of relations, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras, which were in turn an extension of Koszul algebras introduced by S. Priddy. Our definition is in some sense as closest as possible to the one given in the homogeneous case. Indeed, we give an equivalent description of the new definition in terms of the Tor (or Ext) groups, similar to the existing one for homogeneous algebras, and also a complete characterization of the multi-Koszul property, which derives from the study of some associated homogeneous algebras, providing a very strong link between the new definition and the generalized Koszul property for the associated homogeneous algebras mentioned before. We further obtain an explicit description of the Yoneda algebra of a multi-Koszul algebra. As a consequence, we get that the Yoneda algebra of a multi-Koszul algebra is generated in degrees 1 and 2, so a K2 algebra in the sense of T. Cassidy and B. Shelton. We also exhibit several examples and we provide a minimal graded projective resolution of the algebra A considered as an A-bimodule, which may be used to compute the Hochschild (co)homology groups. Finally, we find necessary and sufficient conditions on some (fixed) sequences of vector subspaces of the tensor powers of the base space V to obtain in this case the multi-Koszul property in the case we have relations in only two degrees.
Fil: Herscovich Ramoneda, Estanislao Benito. Universite Joseph Fourier; Francia. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Rey, Andrea. Universidad de San Andrés; Argentina
Materia
Homological Algebra
Koszul Algebras
Yoneda Algebra
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/83269

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spelling On a definition of multi-Koszul algebrasHerscovich Ramoneda, Estanislao BenitoRey, AndreaHomological AlgebraKoszul AlgebrasYoneda Algebrahttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In this article we introduce the notion of multi-Koszul algebra for the case of a nonnegatively graded connected algebra with a finite number of generators of degree 1 and with a finite number of relations, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras, which were in turn an extension of Koszul algebras introduced by S. Priddy. Our definition is in some sense as closest as possible to the one given in the homogeneous case. Indeed, we give an equivalent description of the new definition in terms of the Tor (or Ext) groups, similar to the existing one for homogeneous algebras, and also a complete characterization of the multi-Koszul property, which derives from the study of some associated homogeneous algebras, providing a very strong link between the new definition and the generalized Koszul property for the associated homogeneous algebras mentioned before. We further obtain an explicit description of the Yoneda algebra of a multi-Koszul algebra. As a consequence, we get that the Yoneda algebra of a multi-Koszul algebra is generated in degrees 1 and 2, so a K2 algebra in the sense of T. Cassidy and B. Shelton. We also exhibit several examples and we provide a minimal graded projective resolution of the algebra A considered as an A-bimodule, which may be used to compute the Hochschild (co)homology groups. Finally, we find necessary and sufficient conditions on some (fixed) sequences of vector subspaces of the tensor powers of the base space V to obtain in this case the multi-Koszul property in the case we have relations in only two degrees.Fil: Herscovich Ramoneda, Estanislao Benito. Universite Joseph Fourier; Francia. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Rey, Andrea. Universidad de San Andrés; ArgentinaAcademic Press Inc Elsevier Science2013-02info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/83269Herscovich Ramoneda, Estanislao Benito; Rey, Andrea; On a definition of multi-Koszul algebras; Academic Press Inc Elsevier Science; Journal of Algebra; 376; 2-2013; 196-2270021-8693CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.1016/j.jalgebra.2012.11.030info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0021869312005959info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:01:02Zoai:ri.conicet.gov.ar:11336/83269instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:01:02.751CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv On a definition of multi-Koszul algebras
title On a definition of multi-Koszul algebras
spellingShingle On a definition of multi-Koszul algebras
Herscovich Ramoneda, Estanislao Benito
Homological Algebra
Koszul Algebras
Yoneda Algebra
title_short On a definition of multi-Koszul algebras
title_full On a definition of multi-Koszul algebras
title_fullStr On a definition of multi-Koszul algebras
title_full_unstemmed On a definition of multi-Koszul algebras
title_sort On a definition of multi-Koszul algebras
dc.creator.none.fl_str_mv Herscovich Ramoneda, Estanislao Benito
Rey, Andrea
author Herscovich Ramoneda, Estanislao Benito
author_facet Herscovich Ramoneda, Estanislao Benito
Rey, Andrea
author_role author
author2 Rey, Andrea
author2_role author
dc.subject.none.fl_str_mv Homological Algebra
Koszul Algebras
Yoneda Algebra
topic Homological Algebra
Koszul Algebras
Yoneda Algebra
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv In this article we introduce the notion of multi-Koszul algebra for the case of a nonnegatively graded connected algebra with a finite number of generators of degree 1 and with a finite number of relations, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras, which were in turn an extension of Koszul algebras introduced by S. Priddy. Our definition is in some sense as closest as possible to the one given in the homogeneous case. Indeed, we give an equivalent description of the new definition in terms of the Tor (or Ext) groups, similar to the existing one for homogeneous algebras, and also a complete characterization of the multi-Koszul property, which derives from the study of some associated homogeneous algebras, providing a very strong link between the new definition and the generalized Koszul property for the associated homogeneous algebras mentioned before. We further obtain an explicit description of the Yoneda algebra of a multi-Koszul algebra. As a consequence, we get that the Yoneda algebra of a multi-Koszul algebra is generated in degrees 1 and 2, so a K2 algebra in the sense of T. Cassidy and B. Shelton. We also exhibit several examples and we provide a minimal graded projective resolution of the algebra A considered as an A-bimodule, which may be used to compute the Hochschild (co)homology groups. Finally, we find necessary and sufficient conditions on some (fixed) sequences of vector subspaces of the tensor powers of the base space V to obtain in this case the multi-Koszul property in the case we have relations in only two degrees.
Fil: Herscovich Ramoneda, Estanislao Benito. Universite Joseph Fourier; Francia. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Rey, Andrea. Universidad de San Andrés; Argentina
description In this article we introduce the notion of multi-Koszul algebra for the case of a nonnegatively graded connected algebra with a finite number of generators of degree 1 and with a finite number of relations, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras, which were in turn an extension of Koszul algebras introduced by S. Priddy. Our definition is in some sense as closest as possible to the one given in the homogeneous case. Indeed, we give an equivalent description of the new definition in terms of the Tor (or Ext) groups, similar to the existing one for homogeneous algebras, and also a complete characterization of the multi-Koszul property, which derives from the study of some associated homogeneous algebras, providing a very strong link between the new definition and the generalized Koszul property for the associated homogeneous algebras mentioned before. We further obtain an explicit description of the Yoneda algebra of a multi-Koszul algebra. As a consequence, we get that the Yoneda algebra of a multi-Koszul algebra is generated in degrees 1 and 2, so a K2 algebra in the sense of T. Cassidy and B. Shelton. We also exhibit several examples and we provide a minimal graded projective resolution of the algebra A considered as an A-bimodule, which may be used to compute the Hochschild (co)homology groups. Finally, we find necessary and sufficient conditions on some (fixed) sequences of vector subspaces of the tensor powers of the base space V to obtain in this case the multi-Koszul property in the case we have relations in only two degrees.
publishDate 2013
dc.date.none.fl_str_mv 2013-02
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/83269
Herscovich Ramoneda, Estanislao Benito; Rey, Andrea; On a definition of multi-Koszul algebras; Academic Press Inc Elsevier Science; Journal of Algebra; 376; 2-2013; 196-227
0021-8693
CONICET Digital
CONICET
url http://hdl.handle.net/11336/83269
identifier_str_mv Herscovich Ramoneda, Estanislao Benito; Rey, Andrea; On a definition of multi-Koszul algebras; Academic Press Inc Elsevier Science; Journal of Algebra; 376; 2-2013; 196-227
0021-8693
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jalgebra.2012.11.030
info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0021869312005959
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Academic Press Inc Elsevier Science
publisher.none.fl_str_mv Academic Press Inc Elsevier Science
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
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instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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