Construction of stable equivalences of Morita type for finitedimensional algebras I
Authors:
Yuming Liu and Changchang Xi
Journal:
Trans. Amer. Math. Soc. 358 (2006), 25372560
MSC (2000):
Primary 16G10, 16E30; Secondary 16G70, 18G05, 20J05
Published electronically:
September 22, 2005
MathSciNet review:
2204043
Fulltext PDF Free Access
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Abstract: In the representation theory of finite groups, the stable equivalence of Morita type plays an important role. For general finitedimensional algebras, this notion is still of particular interest. However, except for the class of selfinjective algebras, one does not know much on the existence of such equivalences between two finitedimensional algebras; in fact, even a nontrivial example is not known. In this paper, we provide two methods to produce new stable equivalences of Morita type from given ones. The main results are Corollary 1.2 and Theorem 1.3. Here the algebras considered are not necessarily selfinjective. As a consequence of our constructions, we give an example of a stable equivalence of Morita type between two algebras of global dimension , such that one of them is quasihereditary and the other is not. This shows that stable equivalences of Morita type do not preserve the quasiheredity of algebras. As another byproduct, we construct a Morita equivalence inside each given stable equivalence of Morita type between algebras and . This leads not only to a general formulation of a result by Linckelmann (1996), but also to a nice correspondence of some torsion pairs in mod with those in mod if both and are symmetric algebras. Moreover, under the assumption of symmetric algebras we can get a new stable equivalence of Morita type. Finally, we point out that stable equivalences of Morita type are preserved under separable extensions of ground fields.
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Additional Information
Yuming Liu
Affiliation:
School of Mathematical Sciences, Beijing Normal University, 100875 Beijing, People’s Republic of China
Email:
liuym2@263.net
Changchang Xi
Affiliation:
School of Mathematical Sciences, Beijing Normal University, 100875 Beijing, People’s Republic of China
Email:
xicc@bnu.edu.cn
DOI:
http://dx.doi.org/10.1090/S000299470503775X
PII:
S 00029947(05)03775X
Received by editor(s):
July 28, 2003
Received by editor(s) in revised form:
June 18, 2004
Published electronically:
September 22, 2005
Article copyright:
© Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
