Monic monomial representations I Gorenstein-projective modules

For a $k$-algebra $A$, a quiver $Q$, and an ideal $I$ of $kQ$ generated by monomial relations, let $\mathrm{\Lambda }:=A{\otimes }_{k}kQ\mathrm{/}I$. We introduce the monic representations of $(Q,I)$ over $A$. We give properties of the structural maps of monic representations, and prove that the category $\mathrm{m}\mathrm{o}\mathrm{n}(Q,I,A)$ of the monic representations of $(Q,I)$ over $A$ is a resolving subcategory of $\mathrm{r}\mathrm{e}\mathrm{p}(Q,I,A)$. We introduce the condition $(\mathrm{G})$. The main result claims that a $\text{m}$-module is Gorenstein-projective if and only if it is a monic module satisfying $(\mathrm{G})$. As consequences, the monic $\text{m}$-modules are exactly the projective $\text{m}$-modules if and only if $A$ is semisimple; and they are exactly the Gorenstein-projective $\text{m}$-modules if and only if $A$ is selfinjective, and if and only if $\mathrm{m}\mathrm{o}\mathrm{n}(Q,I,A)$ is Frobenius.