Representations of Quivers over Artinian Rings

Abstract

Given a unitary artinian ring R and a finite acyclic quiver Q, let Λ := RQ be the path ring of Q over R. Then Gorenstein-projective Λ-modules are exactly the separated monic representations of Q over R which satisfy the local Gorenstein-projective condition. We denote by smon(Q,R) the category of all finitely generated separated monic representations of Q over R, and $\mathcal {G}p({\varLambda })$ the category of all finitely generated Gorenstein-projective Λ-modules. If R is a selfinjective ring, then $\mathcal {G}p({\varLambda })=\text {smon}{\it (Q, R) }$ . As an application, if R is a commutative uniserial selfinjective ring of length 2 (here it means that as a regular module R is uniserial with length 2), let 0 ≠ a ∈radR and $\bar {R}=R/\text {rad}\textit {R}$ , our main result says that there is a full functor $H: \mathcal {G}p({\varLambda }) \to \text {mod} {\bar {R}Q}$ which induces a bijection between the indecomposable non-projetive Gorenstein-projective Λ-modules and the indecomposable $ \bar {R}Q$ -modules. Moreover, in this case, each Gorenstein-projective Λ-module is strongly Gorenstein-projective.