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 for some positive integer d. In particular, we classify the Auslander-Reiten (=AR) quivers of CMA for a large class of d-self-injective dg algebras A in terms of (−d)-Calabi-Yau (=CY) configurations, which are Riedtmann's configurations 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 CMA is given by C. For type An, by using a bijection between (−d)-CY configurations and certain purely combinatorial objects which we call maximal d-Brauer relations given by Coelho Simões, we construct such A through a Brauer tree dg algebra.