On string algebras and the Cohen-Macaulay Auslander algebras

Abstract

The Cohen-Macaulay Auslander algebra of an algebra $A$ is defined as the endomorphism algebra of the direct sum of all indecomposable Gorenstein projective $A$-modules. The Cohen-Macaulay Auslander algebra of any string algebra is explicitly constructed in this paper.% and describe some properties of a class of special string algebras by their Cohen-Macaulay Auslander algebras. Moreover, it is shown that a class of special string algebras, which are called to be string algebras with {\it G-condition}, are representation-finite if and only if their Cohen-Macaulay Auslander algebras are representation-finite. As applications, the derived discreteness and some homological dimension of gentle algebras are described in terms of their Cohen-Macaulay Auslander algebras. MSC2020: 16G10, 16G70, 16P10.