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

DOI:
https://doi.org/10.1090/S0002-9947-05-03775-X

Published electronically:
September 22, 2005

MathSciNet review:
2204043

Full-text PDF

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 , 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 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.

**1.**M. AUSLANDER,*Representation dimension of Artin algebras*. Queen Mary College Mathematics Notes, Queen Mary College, London, 1971.**2.**M. AUSLANDER, I. REITEN AND S.O. SMALø,*Representation theory of Artin algebras*. Cambridge University Press, 1995. MR**1314422 (96c:16015)****3.**M. BROU´E, Equivalences of blocks of group algberas. In:*Finite dimensional algebras and related topics.*V. Dlab and L.L. Scott(eds.), Kluwer, 1994, 1-26. MR**1308978 (97c:20004)****4.**H. CARTAN AND S. EILENBERG,*Homological algebra.*Princeton Landmarks in Mathematics, 1973. Originally published in 1956. MR**1731415 (2000h:18022)****5.**Y.A. DROZD AND V.V. KIRICHENKO,*Finite dimensional algebras.*Springer-Verlag, Berlin, 1994. MR**1284468 (95i:16001)****6.**H. KRAUSE, Representation type and stable equivalences of Morita type for finite dimensional algebras. Math. Zeit. 229(1998), 601-606. MR**1664779 (99k:16024)****7.**M. LINCKELMANN, Stable equivalences of Morita type for selfinjective algebras and p-groups. Math. Zeit. 223(1996), 87-100. MR**1408864 (97j:20011)****8.**Y.M. LIU, On stable equivalences of Morita type for finite dimensional algebras. Proc. Amer. Math. Soc. 131(2003), 2657-2662. MR**1974320 (2004a:16021)****9.**Y.M. LIU AND C.C. XI, Construction of stable equivalences of Morita type for finite dimensional algebras II. Preprint, 2004, available at: http://math.bnu.edu.cn/ccxi/.**10.**I. REITEN, Stable equivalence of self-injective algebras. J. Algebra. 40(1976), 63-74.MR**0409560 (53:13314)****11.**J. RICKARD, Derived equivalences as derived functors. J. London Math. Soc. 43(1991), 37-48.MR**1099084 (92b:16043)****12.**J. RICKARD, Some recent advances in modular representation theory. Canad. Math. Soc. Conf. Proc. 23 (1998), 157-178. MR**1648606 (99h:20011)****13.**J. RICKARD, Equivalences of derived categories for symmetric algebras. J. Algebra 257(2002), 460-481. MR**1947972 (2004a:16023)****14.**J. ROTMAN,*An introduction to homological algebra*. Academic Press, New York, San Francisco, London, 1979. MR**0538169 (80k:18001)****15.**C.C. XI, Representation dimension and quasi-hereditary algebras. Adv. in Math. 168(2002), 193-212. MR**1912131 (2003h:16015)****16.**C.C. XI, On the finitistic dimension conjecture II. Related to finite global dimension. Adv. in Math. (to appear). Preprint is available at: http://math.bnu.edu.cn/ccxi/.

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:
https://doi.org/10.1090/S0002-9947-05-03775-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.