favicon Frontiers in Mathematical Sciences
7th Conference
University of Isfahan - January 1-3, 2020
Cohen-Macaulay artin algebras of finite Cohen-Macaulay type
Abdolnaser Bahlekeh, Gonbad Kavous University
Date, Time, and Venue:  Thursday, January 2 | 11:45-12:30 | Hall 2
The pure-semisimple conjecture predicts that every left pure-semisimple ring (a ring over which ev- ery left module is a direct sum of finitely generated ones) is of finite representation type. Left pure- semisimple rings are known to be left artinian by a result of Chase [6, Theorem 4.4]. The validity of the pure-semisimple conjecture for artin algebras comes from a famous result of Auslander [1] (also Ringel- Tachikawa [7]) where they have shown that an artin algebra $ \Lambda $ is of finite representation type if and only if every left $ \Lambda $-module is a direct sum of finitely generated modules. Recall that an artin algebra $ \Lambda $ is of finite representation type, provided that the set of isomorphism classes of indecomposable finitely generated modules is finite. Inspired by Auslander's result, Chen [5] conjectured that Auslander-type result should be true for Gorenstein projective modules: an artin algebra $ \Lambda $ is of finite Cohen-Macaulay type, in the sense that there are only finitely many isomorphism classes of indecomposable finitely generated Gorenstein projective $ \Lambda $-modules, if and only if any left Gorenstein projective $ \Lambda $-module is a direct sum of finitely generated modules. This conjecture has been answered affirmatively by Chen [5] for Gorenstein artin algberas, and by Beligiannis [4] for virtually Gorenstein artin algebras. In this talk, which is based on a joint work with Shokrollah Salarian and Fahimeh Sadat Fotouhi, we will examine the validity of this conjecture for Cohen-Macaulay artin algebras. This notion, which is a generalization of Gorenstein artin algebras, has been introduced by Auslander and Reiten; see [2, 3]. Recall that an artin algebra $ \Lambda $ is said to be a Cohen-Macaulay algebra, if there is a $ \Lambda $-bimodule $\omega$ such that the pair of adjoint functors $(\omega\otimes_{\Lambda}-,\hbox{Hom}_{\Lambda} (\omega,-)) $ induces mutually inverse equivalences between the full subcategories of finitely generated $ \Lambda $-modules, mod $ \Lambda $, consisting of the $ \Lambda $-modules of finite injective dimension and the $ \Lambda $-modules of finite projective dimension.

[1] M. Auslander, A functorial approach to representation theory, in Representatios of Algebra, Workshop Notes of the Third Inter. Confer., Lecture Notes Math. 944, 105-179, Springer-Verlag, 1982.
[2] M. Auslander and I. Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), no. 1, 111-152.
[3] M. Auslander and I. Reiten, Cohen-Macaulay and Gorenstein artin algebras, in Representation theory of finite groups and finite-dimensional algebras (Bielefeld 1991), Progress in mathematics, 95 (eds G. O. Michler and C. M. Ringel) (Birkhauser, Basel, 1991), pp.221-245.
[4] A. Beligiannis, On algebras of finite Cohen-Macaulay type, Adv. Math. 226 (2011), no. 2, 1973-2019.
[5] X. W. Chen, An Auslander-type result for Gorenstein projective modules, Adv. Math. 218 (2008), 2043-2050.
[6] S. U. Chase, Direct product of modules, Trans. Amer. Math. Soc. 97 (1960), 457-473.
[7] C. M. Ringel and H. Tachikawa, QF-3 rings, J. Reine Angew. Math. 272 (1975), 49-72.
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