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Derived equivalences and stable equivalences of Morita type, I

Published online by Cambridge University Press:  11 January 2016

Wei Hu  and
Changchang Xi
Wei Hu
Affiliation:
School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, 100875 Beijing, People’s Republic of China, huwei@bnu.edu.cn
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Abstract

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For self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this article, we first show that each derived equivalence F between the derived categories of Artin algebras A and B arises naturally as a functor between their stable module categories, which can be used to compare certain homological dimensions of A with that of B. We then give a sufficient condition for the functor to be an equivalence. Moreover, if we work with finite-dimensional algebras over a field, then the sufficient condition guarantees the existence of a stable equivalence of Morita type. In this way, we extend the classical result of Rickard. Furthermore, we provide several inductive methods for constructing those derived equivalences that induce stable equivalences of Morita type. It turns out that we may produce a lot of (usually not self-injective) finite-dimensional algebras that are both derived-equivalent and stably equivalent of Morita type; thus, they share many common invariants.

Keywords

18E3016G1018G2016D90
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 200 , December 2010 , pp. 107 - 152
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2010

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