  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.
