Monic monomial representations I Gorenstein-projective modules

Abstract

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

Cited by in corpus (3)

• Singularity categories of Gorenstein monomial algebras
• A description of Gorenstein projective modules over the tensor products of algebras
• Singularity categories of representations of algebras over local rings