The puresemisimple conjecture predicts that every left puresemisimple 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
puresemisimple 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 Auslandertype result
should be true for Gorenstein projective modules: an artin algebra $ \Lambda $ is of finite CohenMacaulay 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 CohenMacaulay 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 CohenMacaulay 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.
References
[1] M. Auslander, A functorial approach to representation theory, in Representatios of Algebra, Workshop Notes of
the Third Inter. Confer., Lecture Notes Math. 944, 105179, SpringerVerlag, 1982.
[2] M. Auslander and I. Reiten, Applications of contravariantly finite subcategories, Adv. Math. 86 (1991), no. 1,
111152.
[3] M. Auslander and I. Reiten, CohenMacaulay and Gorenstein artin algebras, in Representation theory of finite
groups and finitedimensional algebras (Bielefeld 1991), Progress in mathematics, 95 (eds G. O. Michler and C. M.
Ringel) (Birkhauser, Basel, 1991), pp.221245.
[4] A. Beligiannis, On algebras of finite CohenMacaulay type, Adv. Math. 226 (2011), no. 2, 19732019.
[5] X. W. Chen, An Auslandertype result for Gorenstein projective modules, Adv. Math. 218 (2008), 20432050.
[6] S. U. Chase, Direct product of modules, Trans. Amer. Math. Soc. 97 (1960), 457473.
[7] C. M. Ringel and H. Tachikawa, QF3 rings, J. Reine Angew. Math. 272 (1975), 4972.
