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Stable equivalences of adjoint type

Stable equivalences of adjoint type In this paper we define a class of stable equivalences, namely, the stable equivalences of adjoint type, and study the Hochschild cohomology groups of algebras that are linked by a stable equivalence of adjoint type. This notion of adjoint type is a special case of Morita type, covers the stable equivalence of Morita type for self-injective algebras, and thus includes the case where Broué's conjecture was made (see for instance Broué M.: Equivalences of blocks of group algberas. In: Finite dimensional algebras and related topics (V. Dlab and L. L. Scott eds.). Kluwer, 1994, 1–26). The main results in this paper are: Let A and B be two artin k -algebras such that A and B are projective over k , and let H n ( A ) and H n ( B ) be the n -th Hochschild cohomology groups of A and B , respectively. (1) If A and B are stably equivalent of adjoint type, then H n ( A ) ≃ H n ( B ) for all n ≥ 1. (2) If A and B are stably equivalent of Morita type, then the absolute values of Cartan determinants of A and B are equal. In particular, two cellular algebras over a field have the same Cartan determinant if they are stably equivalent of Morita type. 2000 Mathematics Subject Classification: 16G10, 16E30; 16G70, 18G05, 20J05. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Stable equivalences of adjoint type

Xi, Changchang
Forum Mathematicum , Volume 20 (1) – Jan 1, 2008
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Publisher
de Gruyter
Copyright
© Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/FORUM.2008.004
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Abstract

In this paper we define a class of stable equivalences, namely, the stable equivalences of adjoint type, and study the Hochschild cohomology groups of algebras that are linked by a stable equivalence of adjoint type. This notion of adjoint type is a special case of Morita type, covers the stable equivalence of Morita type for self-injective algebras, and thus includes the case where Broué's conjecture was made (see for instance Broué M.: Equivalences of blocks of group algberas. In: Finite dimensional algebras and related topics (V. Dlab and L. L. Scott eds.). Kluwer, 1994, 1–26). The main results in this paper are: Let A and B be two artin k -algebras such that A and B are projective over k , and let H n ( A ) and H n ( B ) be the n -th Hochschild cohomology groups of A and B , respectively. (1) If A and B are stably equivalent of adjoint type, then H n ( A ) ≃ H n ( B ) for all n ≥ 1. (2) If A and B are stably equivalent of Morita type, then the absolute values of Cartan determinants of A and B are equal. In particular, two cellular algebras over a field have the same Cartan determinant if they are stably equivalent of Morita type. 2000 Mathematics Subject Classification: 16G10, 16E30; 16G70, 18G05, 20J05.

Journal

Forum Mathematicum – de Gruyter

Published: Jan 1, 2008

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