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Projective bimodule resolutions of an algebra and vanishing of the second Hochschild cohomology group

Projective bimodule resolutions of an algebra and vanishing of the second Hochschild cohomology... Abstract. In this paper we construct explicitly the first terms in the minimal projective bimodule resolution of a finite-dimensional algebra from the minimal right resolution of each of the simple modules. This result is used to give vanishing results for HH 2 of a finite-dimensional algebra, and in particular shows that HH 2 ¼ 0 for all Mobius algebras, with the exception of the pre¨ projective algebra of type A3 . 2000 Mathematics Subject Classification: 16E05, 16E40. 1 Introduction Happel [9] provided a description of each of the projective modules occurring in a minimal projective bimodule resolution for a finite-dimensional algebra L, given a minimal projective resolution of each of the right simple L-modules. The aim of this paper is to give an explicit matrix construction of the first three maps in this minimal projective bimodule resolution of L, using the minimal projective resolution of each of the right simple L-modules as given in [8]. As an application we give some vanishing results for HH 2 ðLÞ and specifically show that HH 2 ðLÞ ¼ 0 for all Mobius algebras L, with the one exception of the prepro¨ jective algebra of type A3 where it is already known http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Projective bimodule resolutions of an algebra and vanishing of the second Hochschild cohomology group

Green, Edward L.; Snashall, Nicole
Forum Mathematicum , Volume 16 (1) – Jan 13, 2004
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References (8)

Publisher
de Gruyter
Copyright
Copyright © 2004 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2004.003
Publisher site
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Abstract

Abstract. In this paper we construct explicitly the first terms in the minimal projective bimodule resolution of a finite-dimensional algebra from the minimal right resolution of each of the simple modules. This result is used to give vanishing results for HH 2 of a finite-dimensional algebra, and in particular shows that HH 2 ¼ 0 for all Mobius algebras, with the exception of the pre¨ projective algebra of type A3 . 2000 Mathematics Subject Classification: 16E05, 16E40. 1 Introduction Happel [9] provided a description of each of the projective modules occurring in a minimal projective bimodule resolution for a finite-dimensional algebra L, given a minimal projective resolution of each of the right simple L-modules. The aim of this paper is to give an explicit matrix construction of the first three maps in this minimal projective bimodule resolution of L, using the minimal projective resolution of each of the right simple L-modules as given in [8]. As an application we give some vanishing results for HH 2 ðLÞ and specifically show that HH 2 ðLÞ ¼ 0 for all Mobius algebras L, with the one exception of the prepro¨ jective algebra of type A3 where it is already known

Journal

Forum Mathematicum – de Gruyter

Published: Jan 13, 2004

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