Cohen-Macaulay differential graded modules and negative Calabi-Yau configurations

Abstract

In this paper, we introduce the class of Cohen-Macaulay (=CM) dg (=differential graded) modules over Gorenstein dg algebras and study their basic properties. We show that the category of CM dg modules forms a Frobenius extriangulated category, in the sense of Nakaoka and Palu, and it admits almost split extensions. We also study representation-finite $d$-self-injective dg algebras $A$ in detail. In particular, we classify the Auslander-Reiten (=AR) quivers of CM $A$ for those $A$ in terms of $(-d)$-Calabi-Yau (=CY) configurations, which are Riedtmann's configuration for the case $d=1$. For any given $(-d)$-CY configuration $C$, we show there exists a $d$-self-injective dg algebra $A$, such that the AR quiver of CM $A$ is given by $C$. For type $A_{n}$, by using a bijection between $(-d)$-CY configurations and certain purely combinatorial objects which we call maximal $d$-Brauer relations given by Coelho Sim\~oes, we construct such $A$ through a Brauer tree dg algebra.