Strongly Gorenstein-projective Quiver Representations

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 >= 2, let A = k[x]/< x(t)>, each Gorenstein-projective AQ-module is strongly Gorensteinprojective if and only if A = k[x]/< x(2)>.