On algebras of finite Cohen-Macaulay type

By Apostolos Beligiannis. 2011

126 citations recorded by the source corpus.

Abstract We study Artin algebras Λ and commutative Noetherian complete local rings R in connection with the following decomposition property of Gorenstein-projective modules: (†) any Gorenstein-projective module is a direct sum of finitely generated modules . We show that the class of algebras Λ enjoying ( † ) coincides with the class of virtually Gorenstein algebras of finite Cohen–Macaulay type, introduced in Beligiannis and Reiten (2007) [27] , Beligiannis (2005) [24] . Thus we solve the problem stated in Chen (2008) [33] . This is proved by characterizing when a resolving subcategory is of finite representation type in terms of decomposition properties of its closure under filtered colimits, thus generalizing a classical result of Auslander (1976) [9] and Ringel and Tachikawa (1974) [63] . In the commutative case, if R admits a non-free finitely generated Gorenstein-projective module, then we show that R is of finite Cohen–Macaulay type iff R is Gorenstein and satisfies ( † ). We also generalize a result of Yoshino (2005) [68] by characterizing when finitely generated modules without extensions with the ring are Gorenstein-projective. Finally we study the (stable) relative Auslander algebra of a virtually Gorenstein algebra of finite Cohen–Macaulay type and, under the presence of a cluster tilting object, we give descriptions of the stable category of Gorenstein-projective modules in terms of the cluster category associated to the quiver of the stable relative Auslander algebra. In this setting we show that the cluster category is invariant under derived equivalences.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ddc80e293aad9a04adeaa8269181ed00cb1d2f09

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings (2015, background): “In fact, there are connections with classification problems (Ringel, Schmidmeier [40–42], Xiong, Zhang, Zhang [45]), with weighted projective lines (Kussin, Lenzing, Meltzer [31]), and with aspects of Gorenstein homological algebra (Beligiannis [10, 11], Zhang [49], Chen [17]).”
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].”
The Stable Monomorphism Category of a Frobenius category (2009, methodology): “The case that the ring is k[t]/(t) with k a field is studied by Simson ([37]) and also by Beligiannis ([7]).”
On Artin Algebras Arising from Morita Contexts (2013, background): “Finally recall from [11], [12] that an Artin algebra Λ is said to be of finite Cohen-Macaulay type if the category GprojΛ of finitely generated Gorenstein-projective Λ-modules is of finite representation type, i.”