Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Construction of stable equivalences of Morita type for finite-dimensional algebras I


Authors: Yuming Liu and Changchang Xi
Journal: Trans. Amer. Math. Soc. 358 (2006), 2537-2560
MSC (2000): Primary 16G10, 16E30; Secondary 16G70, 18G05, 20J05
Published electronically: September 22, 2005
MathSciNet review: 2204043
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In the representation theory of finite groups, the stable equivalence of Morita type plays an important role. For general finite-dimensional algebras, this notion is still of particular interest. However, except for the class of self-injective algebras, one does not know much on the existence of such equivalences between two finite-dimensional algebras; in fact, even a non-trivial 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 self-injective. As a consequence of our constructions, we give an example of a stable equivalence of Morita type between two algebras of global dimension $4$, such that one of them is quasi-hereditary and the other is not. This shows that stable equivalences of Morita type do not preserve the quasi-heredity of algebras. As another by-product, we construct a Morita equivalence inside each given stable equivalence of Morita type between algebras $A$ and $B$. 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 $A$-mod with those in $B$-mod if both $A$ and $B$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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 16G10, 16E30, 16G70, 18G05, 20J05

Retrieve articles in all journals with MSC (2000): 16G10, 16E30, 16G70, 18G05, 20J05


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/S0002-9947-05-03775-X
PII: S 0002-9947(05)03775-X
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.