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.

中文翻译：

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Cohen-Macaulay 差分分级模块和负 Calabi-Yau 配置

在本文中，我们介绍了 Gorenstein dg 代数上的 Cohen-Macaulay (=CM) dg（=微分分级）模块的类并研究了它们的基本性质。我们证明了 CM dg 模块的范畴形成了 Frobenius 外三角化范畴，在 Nakaoka 和 Palu 的意义上，它承认几乎分裂的扩展。我们还详细研究了表示有限$d$-self-injective dg 代数$A$。特别地，我们根据 $(-d)$-Calabi-Yau (=CY) 配置对那些 $A$ 的 CM $A$ 的 Auslander-Reiten (=AR) 颤动进行分类，这是 Riedtmann 对案例的配置$d=1$。对于任何给定的 $(-d)$-CY 配置 $C$，我们证明存在一个 $d$-self-injective dg 代数 $A$，使得 CM $A$ 的 AR quiver 由 $C$ 给出. 对于 $A_{n}$ 类型，