Total Acyclicity for Complexes of Representations of Quivers

Abstract

We study total acyclicity for complexes of projective and injective representations of quivers. A classification is given for such complexes in terms of associated vertex-complexes. When the base ring or the quiver is nice enough, this classification is used to prove the existence of Gorenstein projective precovers in the category of representations of quivers. Furthermore, we exploit this local description to obtain some criteria for the category of representations of a quiver to be Gorenstein or virtually Gorenstein. If Λ is an artin algebra it is proved that, for an arbitrary quiver 𝒬, the representation category Rep(𝒬, Λ) is virtually Gorenstein whenever Λ is virtually Gorenstein. A description of Gorenstein projective and Gorenstein injective representations of quivers over general rings is also provided.