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 , 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.**Maurice Auslander, Idun Reiten, and Sverre O. Smalø,*Representation theory of Artin algebras*, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR**1314422****3.**Michel Broué,*Equivalences of blocks of group algebras*, Finite-dimensional algebras and related topics (Ottawa, ON, 1992) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, pp. 1–26. MR**1308978**, 10.1007/978-94-017-1556-0_1**4.**Henri Cartan and Samuel Eilenberg,*Homological algebra*, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1999. With an appendix by David A. Buchsbaum; Reprint of the 1956 original. MR**1731415****5.**Yurij A. Drozd and Vladimir V. Kirichenko,*Finite-dimensional algebras*, Springer-Verlag, Berlin, 1994. Translated from the 1980 Russian original and with an appendix by Vlastimil Dlab. MR**1284468****6.**Henning Krause,*Representation type and stable equivalence of Morita type for finite-dimensional algebras*, Math. Z.**229**(1998), no. 4, 601–606. MR**1664779**, 10.1007/PL00004671**7.**Markus Linckelmann,*Stable equivalences of Morita type for self-injective algebras and 𝑝-groups*, Math. Z.**223**(1996), no. 1, 87–100. MR**1408864**, 10.1007/PL00004556**8.**Yuming Liu,*On stable equivalences of Morita type for finite dimensional algebras*, Proc. Amer. Math. Soc.**131**(2003), no. 9, 2657–2662 (electronic). MR**1974320**, 10.1090/S0002-9939-03-06831-X**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.**Idun Reiten,*Stable equivalence of self-injective algebras*, J. Algebra**40**(1976), no. 1, 63–74. MR**0409560****11.**Jeremy Rickard,*Derived equivalences as derived functors*, J. London Math. Soc. (2)**43**(1991), no. 1, 37–48. MR**1099084**, 10.1112/jlms/s2-43.1.37**12.**Jeremy Rickard,*Some recent advances in modular representation theory*, Algebras and modules, I (Trondheim, 1996) CMS Conf. Proc., vol. 23, Amer. Math. Soc., Providence, RI, 1998, pp. 157–178. MR**1648606****13.**Jeremy Rickard,*Equivalences of derived categories for symmetric algebras*, J. Algebra**257**(2002), no. 2, 460–481. MR**1947972**, 10.1016/S0021-8693(02)00520-3**14.**Joseph J. Rotman,*An introduction to homological algebra*, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR**538169****15.**Changchang Xi,*Representation dimension and quasi-hereditary algebras*, Adv. Math.**168**(2002), no. 2, 193–212. MR**1912131**, 10.1006/aima.2001.2046**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:
http://dx.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.