Gorenstein algebras of finite Cohen–Macaulay type

An Artin algebra A is said to be CM-finite if there are only finitely many isomorphism classes of indecomposable finitely generated Gorenstein-projective A-modules. Inspired by Auslander's idea on representation dimension, we prove that for 2⩽n<∞, A is a CM-finite n-Gorenstein algebra if and only if there is a resolving Gorenstein-projective A-module E such that gl.dimEndAop(E)⩽n.