The stable category of Gorenstein-projective modules over a monomial algebra

Abstract

Let $\Lambda$ be an arbitrary monomial algebra. We investigate the stable category ${\underline{Gproj}}^Z\Lambda$ of graded Gorenstein-projective $\Lambda$-modules and the orbit category ${\underline{Gproj}}^Z\Lambda /\left(1\right)$ induced by ${\underline{Gproj}}^Z\Lambda$ and the degree shift functor (1). We prove that ${\underline{Gproj}}^Z\Lambda$ is triangle equivalent to the bounded derived category of a path algebra of Dynkin type $A$ and that ${\underline{Gproj}}^Z\Lambda /\left(1\right)$ is triangle equivalent to the stable module category of a self-injective Nakayama algebra. Both the path algebra and the self-injective Nakayama algebra will be given explicitly. The latter result provides an explicit description of the stable category of (ungraded) Gorenstein-projective $\Lambda$-modules.