The Cohen-Macaulay Auslander algebras of string algebras

TL;DR

This paper explicitly constructs Cohen-Macaulay Auslander algebras for string algebras, explores their finiteness properties, and characterizes gentle algebras' self-injective dimension through these algebras.

Contribution

It provides an explicit construction of Cohen-Macaulay Auslander algebras for string algebras and links their representation-finiteness to that of the original algebras.

Findings

Cohen-Macaulay Auslander algebras are explicitly constructed for string algebras.

Representation-finiteness of certain string algebras is characterized via their Cohen-Macaulay Auslander algebras.

Self-injective dimension of gentle algebras is characterized using Cohen-Macaulay Auslander algebras.

Abstract

The Cohen-Macaulay Auslander algebra of any string algebra is explicitly constructed in this paper. Furthermore, we show that a class of special string algebras, which are called to be string algebras with G-condition, are representation-finite if and only if their Cohen-Macaulay Auslander algebras are representation-finite. Finally, the self-injective dimension of gentle algebras is characterized using their Cohen-Macaulay Auslander algebras.