Gorenstein injective and projective complexes

In this article we extend the notion of Gorenstein injective and projective modules to that of complexes and characterize such complexes. We prove that over an n-Gorenstein ring every complex has a Gorenstein injective envelope and we show that every such envelope is a quasi-isomorphim.. When the ring is commutative, local and Gorenstein, Auslander announced that every finitely generated R-module has a finitely generated Gorenstein projective cover. We show that every bounded above complex having all terms finitely generated over such a ring has a Gorenstein projective cover and we show that these covers are quasi-isomorphisms.