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

TL;DR

This paper characterizes the stable category of graded Gorenstein-projective modules over a monomial algebra, showing its equivalence to derived categories of Dynkin type A and stable categories of Nakayama algebras, providing explicit descriptions.

Contribution

It establishes explicit triangle equivalences for the stable categories of Gorenstein-projective modules over monomial algebras, linking them to well-understood algebraic structures.

Findings

Stable category is equivalent to the derived category of a Dynkin type A path algebra.

Orbit category is equivalent to the stable module category of a Nakayama algebra.

Provides explicit algebraic descriptions of these categories.

Abstract

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