On algebras of finite Cohen-Macaulay type

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.