2018/12/10 by Minamoto, Hiroyuki, Yamaura, Kota
#FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1812.03746
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 the previous study, 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 \underlineCMℤ A of CM-modules is realized as an admissible subcategory of the derived category Db(mod Λ). As a corollary, we deduce that the Grothendieck group K0(\underlineCMℤ A) is free of finite rank. We give several applications. Among other things, for a path algebra Λ= \Bbbk Q 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.