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 $A = k[x]/\langle x^2 \rangle$.
Luo is supported by the National Science Foundation of China
We sincerely thank the referee for very carefully reading the manuscript and many helpful comments and valuable suggestions which helped improving this paper substantially.
"Strongly Gorenstein-projective Quiver Representations." Taiwanese J. Math. 25 (3) 449 - 461, June, 2021. https://doi.org/10.11650/tjm/201103