An Auslander-Type result for Gorenstein-projective modules

By Xiao-Wu Chen. 2007

62 citations recorded by the source corpus.

An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander's theorem on algebras of finite representation type [M. Auslander, A functorial approach to representation theory, in: Representations of Algebras, Workshop Notes of the Third Internat. Conference, in: Lecture Notes in Math., vol. 944, Springer-Verlag, Berlin, 1982, pp. 105�179; M. Auslander, Representation theory of artin algebras II, Comm. Algebra (1974) 269�310]

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dbf887b93692eaf2494f209bfe366c9c8d1baded

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Citations in the corpus

On algebras of finite Cohen-Macaulay type (2011, methodology): “Thus we solve the problem stated in [35].”
Algebras with radical square zero are either self-injective or CM-free (2011, background): “This problem is closely related to a conjecture stated in [7]; also see [5].”
Gorenstein Homological Algebra of Artin Algebras (2017, background): “An affirmative answer is given for the case where A is Gorenstein ([20]).”