Cohen-Macaulay Auslander algebras of gentle algebras

Abstract

For any gentle algebra $\Lambda=KQ/\langle I\rangle$, following Kalck, we describe the quiver and the relations for its Cohen-Macaulay Auslander algebra $\mathrm{Aus}(\mathrm{Gproj}\Lambda)$ explicitly, and obtain some properties, such as $\Lambda$ is representation-finite if and only if $\mathrm{Aus}(\mathrm{Gproj}\Lambda)$ is; if $Q$ has no loop and any indecomposable $\Lambda$-module is uniquely determined by its dimension vector, then any indecomposable $\mathrm{Aus}(\mathrm{Gproj}\Lambda)$-module is uniquely determined by its dimension vector.