On algebras of finite Cohen-Macaulay type

TL;DR

This paper investigates the structure of Gorenstein-projective modules over Artin algebras and complete local rings, establishing links to finite Cohen-Macaulay type and generalizing classical representation theory results.

Contribution

It generalizes classical results on Gorenstein-projective modules and connects the decomposition property to finite Cohen-Macaulay type in a broad algebraic context.

Findings

Decomposition property relates to (virtually) Gorenstein algebras of finite Cohen-Macaulay type.

Generalization of classical results by Auslander and Ringel-Tachikawa.

Descriptions of stable categories via cluster categories under certain conditions.

Abstract

We study Artin algebras $A$ 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 this direct decomposition property is related to the property of the algebra $A$, or the ring $R$, being (virtually) Gorenstein of finite Cohen-Macaquly type. Along the way we generalize classical results of Auslander and Ringel-Tachikawa from the early seventies, and results of Chen and Yoshino on the structure of Gorenstein-projective modules. Finally we study homological properties of (stable) relative Auslander algebras of virtually Gorenstein algebras 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 suitable cluster categories.