Total Acyclicity for Complexes of Representations of Quivers

Abstract

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. Key Words: Gorenstein projective precoverTotally acyclic complexQuiver representations2010 Mathematics Subject Classification: 16G2018E3018A4055U35 ACKNOWLEDGMENTS The authors are grateful to the anonymous referee for his/her comments on the article. Sh. Salarian thanks the Center of Excellence for Mathematics (University of Isfahan). H. Eshraghi and R. Hafezi thank the Institute for Research in Fundamental Sciences (IPM). The work of the last author is supported in part by a grant from IPM (No. 90130218). Notes Communicated by U. Walther.