Acta mathematica scientia, Series B >
INVARIANTS UNDER STABLE EQUIVALENCES OF MORITA TYPE
Received date: 2009-10-11
Revised date: 2011-01-07
Online published: 2012-03-20
Supported by
Project supported by the National Natural Science Foundation of China (10871170) and the Zhejiang Provincial Natural Science Foundation of China (D7080064). The second author is partially supported by the National Natural Science Foundation of China (10801117).
The aim of this article is to study some invariants of associative algebras under stable equivalences of Morita type. First of all, we show that, if two finite-dimensional self-injective k-algebras are stably equivalent of Morita type, then their orbit algebras are isomorphic. Secondly, it is verified that the quasitilted property of an algebra is invariant under stable equivalences of Morita type. As an application of this result, it is obtained that if an algebra is of finite representation type, then its tilted property is invariant under stable equivalences of Morita type; the other application to partial tilting modules is given in Section 4. Finally, we prove that when two finite-dimensional k-algebras are stably equivalent of Morita type, their repetitive algebras are also stably equivalent of Morita type under certain conditions.
LI Fang , SUN Long-Gang . INVARIANTS UNDER STABLE EQUIVALENCES OF MORITA TYPE[J]. Acta mathematica scientia, Series B, 2012 , 32(2) : 605 -618 . DOI: 10.1016/S0252-9602(12)60042-3
[1] Assem I, Simson D, Skowronski A. Elements of the representation theory of associative algebra. Vol 1. Techniques of representation theory. London Mathematical Society Student Texts, 65. Cambridge: Cambridge University Press, 2006
[2] Dugas A S, Martinez-Villa R. A note on stable equivalence of Morita type. J Pure Appl Algebra, 2007, 208(2): 421–433
[3] Happel D. Triangulated categories in the representation theory of finite-dimensional algebras. London Math Soc Lecture Note Series, Vol 119, 1988
[4] Happel D, Reiten I, Smal´o S O. Tilting in Abelian categories and quasitilted algebras. Mem Amer Math Soc, Vol 575, 1996
[5] Kerner O. Minimal approximations, orbital elementary modules, and orbit algebras of regular modules. J Algebra, 1999, 217: 528–554
[6] Krause H. Representation type and stable equivalence of Morita type for finite-dimensional algebras. Math Z, 1998, 229: 601–606
[7] Lenzing H. Wild Canonical Alebras and Rings of Automorphic Forms//Dlab V, Scott L L. Finite Di-mensional Algebras and Related Topics. NATO ASI Series C Vol 424. Kluwer Academic Press, 1992: 191–212
[8] Liu Y M. On stable equivalences of Morita type for finite-dimensional algebras. Proc Amer Math Soc, 2003, 131: 2657–2662
[9] Liu Y M, Xi C C. Construction of stable equivalences of Morita type for finite-dimensional algebras I. Trans Amer Math Soc, 2005, 358: 2537-2560
[10] Liu Y M, Xi C C. Construction of stable equivalences of Morita type for finite-dimensional algebras II. Math Z, 2005, 251: 21–39
[11] Liu Y M, Xi C C. Construction of stable equivalences of Morita type for finite-dimensional algebras III. J London Math Soc, 2007, 76: 567–585
[12] Martinez-Villa R. Property that are left invariant under stable equivalence. Comm in Algebra, 1990, 18: 4141–4169
[13] Pogorzaly Z. Invariance of Hochschild cohomology algebras under stable equivalences of Morita type. J Math Soc Japan, 2001, 53: 913–918
[14] Pogorzaly Z. Left-right projective bimodules and stable equivalences of Morita type. Colloq Math, 2001, 88: 243–255
[15] Pogorzaly Z. A new invariant of stable equivalence of Morita type. Proc Amer Math Soc, 2002, 131: 343–349
[16] Rickard J. Derived equivalences as derived functors. J London Math Soc, 1991, 43: 37–48
[17] Rotman J J. An Introduction to Homological Algebra. New York: Academic press, 1979
[18] Xi C C. Stable equivalences of adjoint type. Forum Math, 2008, 20: 81–97
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