On the homotopy categories of projective and injective representations of quivers

Abstract

Let R be a ring and Q be a quiver. We study the homotopy categories K(PrjQ) and K(InjQ) consisting, respectively, of projective and injective representations of Q by R-modules. We show that, for certain quivers, these triangulated categories are compactly generated and provide explicit descriptions of compact generating sets. Moreover, in case R is commutative and noetherian with a dualizing complex D, the dualizing functor D⊗R-:K(PrjR) → K(InjR) is extended to a triangulated functor K(PrjQ)→K(InjopQ) which is an equivalence of triangulated categories. This functor, establishes an equivalence on K(PrjQ) and K(InjQ), whenever Q is finite. © 2011 Elsevier Inc.