Monic monomial representations I Gorenstein-projective modules

Xiu-Hua Luo, Pu Zhang

Published 2015-10-17Version 1

For a finite-dimensional algebra $A$ over field $k$, a finite acyclic quiver $Q$, and an ideal $I$ of the path algebra $kQ$ generated by monomial relations, let $\Lambda: = A\otimes_k kQ/I$. We introduce the monic representations of $(Q, I)$ over $A$. By the equivalence $\m\mbox{-}{\rm mod}\cong {\rm rep}(Q, I, A)$, we also have the notion of monic $\m$-module. We give properties of the structural maps of monic representations, and prove that the category of the monic representations of $(Q, I)$ over $A$ is a resolving subcategory of ${\rm rep}(Q, I, A)$. We introduce the condition ${\rm(G)}$. The main result of this paper claims that a $\m$-module is Gorenstein-projective if and only if it is a monic module satisfying ${\rm(G)}$. This provides an inductive construction of Gorenstein-projective modules. As consequences, the monic $\m$-modules are exactly the projective $\m$-modules if and only if $A$ is semisimple, and the monic $\m$-modules are exactly the Gorenstein-projective $\m$-modules if and only if $A$ is selfinjective.