Strongly Gorenstein-projective Quiver Representations

Abstract

Given a field $k$, a finite-dimensional $k$-algebra $A$, and a finite acyclic quiver $Q$, let $AQ$ be the path algebra of $Q$ over $A$. Then the category of representations of $Q$ over $A$ is equivalent to the category of $AQ$-modules. The main result of this paper explicitly describes the strongly Gorenstein-projective $AQ$-modules via the separated monic representations with a local strongly Gorenstein-property. As an application, a necessary and sufficient condition is given on when each Gorenstein-projective $AQ$-module is strongly Gorenstein-projective. As a direct result, for an integer $t \geq 2$, let $A = k[x]/\langle x^t \rangle$, each Gorenstein-projective $AQ$-module is strongly Gorenstein-projective if and only if $A = k[x]/\langle x^2 \rangle$.