On finitely graded Iwanaga-Gorenstein algebras and the stable categories of their (graded) Cohen-Macaulay modules

Abstract

We discuss finitely graded Iwanaga-Gorenstein (IG) algebras $A$ and representation theory of their (graded) Cohen-Macaulay (CM) modules. By quasi-Veronese algebra construction, in principle, we may reduce our study to the case where $A$ is a trivial extension algebra $A = \Lambda \oplus C$ with the grading $ \text{deg} \Lambda = 0, \ \text{deg} C = 1$. In the previous study, we gave a necessary and sufficient condition that $A$ is IG in terms of $\Lambda$ and $C$ by using derived tensor products and derived Homs. For simplicity, we assume that $\Lambda$ is of finite global dimension in the sequel. In this paper, we show that the condition that $A$ is IG, has a triangulated categorical interpretation. We prove that if $A$ is IG, then the graded stable category $\underline{\mathsf{CM}}^{\Bbb{Z}} A$ of CM-modules is realized as an admissible subcategory of the derived category $\mathsf{D}^{\mathrm{b}}(\text{mod } \Lambda)$. As a corollary, we deduce that the Grothendieck group $K_{0}(\underline{\mathsf{CM}}^{\Bbb{Z}} A)$ is free of finite rank. We give several applications. Among other things, for a path algebra $\Lambda= \Bbb{k} Q$ of an $A_{2}$ or $A_{3}$ quiver Q, we give a complete list of $\Lambda$-$\Lambda$-bimodule $C$ such that $\Lambda \oplus C$ is IG (resp. of finite global dimension) by using the triangulated categorical interpretation mentioned above.