An Auslander-type result for Gorenstein-projective modules

TL;DR

This paper establishes a characterization of CM-finite Gorenstein artin algebras, showing that such algebras have all Gorenstein-projective modules decomposable into finitely generated ones, extending Auslander's classical theorem.

Contribution

It provides a new Auslander-type criterion for CM-finiteness in Gorenstein artin algebras, linking module decomposition to finiteness conditions.

Findings

CM-finiteness characterized by decomposability of Gorenstein-projective modules

Extension of Auslander's theorem to Gorenstein context

Finitely generated Gorenstein-projective modules form a key class

Abstract

An artin algebra $A$ is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely generated Gorenstein-projective $A$-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely generated Gorenstein-projective modules. This is an analogue of Auslander's theorem on algebras of finite representation type (\cite{A,A1}).