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

Abstract

Let $\Lambda$ be an arbitrary monomial algebra. We investigate the stable category ${\underline{\mathrm{Gproj}}}^{\mathbb{Z}}\Lambda$ of graded Gorenstein-projective $\Lambda$-modules and the orbit category ${\underline{\mathrm{Gproj}}}^{\mathbb{Z}}\Lambda /(1)$ induced by ${\underline{\mathrm{Gproj}}}^{\mathbb{Z}}\Lambda$ and the degree shift functor $(1)$. We prove that ${\underline{\mathrm{Gproj}}}^{\mathbb{Z}}\Lambda$ is triangle equivalent to the bounded derived category of a path algebra of Dynkin type $\mathbb{A}$ and that ${\underline{\mathrm{Gproj}}}^{\mathbb{Z}}\Lambda /(1)$ 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.

arXiv commentv3: 44 pages, 2 figures, minor improvements to exposition, and corrected typos