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=Λ⊕C with the grading deg⁡Λ=0,deg⁡C=1. In [17] we gave a necessary and sufficient condition that A is IG in terms of Λ and C by using derived tensor products and derived Homs. For simplicity, we assume that Λ 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 CM_ZA of CM-modules is realized as an admissible subcategory of the derived category Db(modΛ). As a corollary, we deduce that the Grothendieck group K0(CM_ZA) is free of finite rank.We give several applications. Among other things, for a path algebra Λ=kQ of an A2 or A3 quiver Q, we give a complete list of Λ-Λ-bimodule C such that Λ⊕C is IG (resp. of finite global dimension) by using the triangulated categorical interpretation mentioned above.