Monic representations and Gorenstein-projective modules

TL;DR

This paper characterizes Gorenstein-projective modules over path algebras of acyclic quivers in terms of monic representations over a finite-dimensional algebra, providing explicit descriptions and criteria for self-injectivity.

Contribution

It introduces the notion of monic representations over a finite-dimensional algebra and explicitly characterizes Gorenstein-projective modules for acyclic quivers, linking them to self-injective algebras.

Findings

Gorenstein-projective modules are explicitly determined via monic representations for acyclic quivers.

A algebra is self-injective if and only if Gorenstein-projective modules coincide with monic representations.

Provides a new perspective on the structure of Gorenstein-projective modules in the context of quiver representations.

Abstract

Let $\Lambda$ be the path algebra of a finite quiver $Q$ over a finite-dimensional algebra $A$. Then $\Lambda$-modules are identified with representations of $Q$ over $A$. This yields the notion of monic representations of $Q$ over $A$. If $Q$ is acyclic, then the Gorenstein-projective $\m$-modules can be explicitly determined via the monic representations. As an application, $A$ is self-injective if and only if the Gorenstein-projective $\m$-modules are exactly the monic representations of $Q$ over $A$.