A reflection equivalence for Gorenstein-projective quiver representations

TL;DR

This paper establishes a reflection equivalence for Gorenstein-projective modules over certain algebras, showing that their stable categories are invariant under orientation changes of the quiver, especially for tree-shaped quivers.

Contribution

It introduces an explicit functorial reflection equivalence for Gorenstein-projective modules at sink vertices, demonstrating invariance of stable categories under quiver orientation changes.

Findings

The functor $F(v)$ is an equivalence of categories.

Stable categories are orientation-independent for tree-shaped quivers.

A symmetry property is verified for type A3 quivers with polynomial algebra.

Abstract

For $\Lambda$ a selfinjective algebra, and $Q$ a finite quiver without oriented cycles, the algebra $\Lambda Q$ is a Gorenstein algebra and the category ${\rm Gproj}\Lambda Q$ of Gorenstein-projective $\Lambda Q$-modules is a Frobenius category. For a sink $v$ of $Q$, we define a functor $F(v) : \underline{\rm Gproj}\Lambda Q\to \underline{\rm Gproj}\Lambda Q(v)$ between the stable categories modulo projectives, where $Q(v)$ is obtained from $Q$ by changing the direction of each arrow ending in $v$. The functor is given by an explicit construction on the level of objects and homomorphisms. Our main result states that $F(v)$ is an equivalence of categories. In the case where the underlying graph of $Q$ is a tree, we deduce that the stable category $\underline{\rm Gproj}\Lambda Q$ does not depend on the orientation of $Q$. Moreover, if $Q$ is a quiver of type $\mathbb A_3$ and $\Lambda=k[T]/(T^n)$ the bounded polynomial algebra, we use the symmetry of the octahedron in the octahedral axiom to verify that the composition of twelve reflections yields the identity on objects.