On algebras of finite Cohen–Macaulay type

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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.