In this article we extend the notion of Gorenstein injective and proje
ctive modules to that of complexes and characterize such complexes. We
prove that over an a-Gorenstein ring every complex has a Gorenstein i
njective envelope and we show that every such envelope is a quasi-isom
orphim. When the ring is commutative, local and Gorenstein, Auslander
announced that every finitely generated R-module has a finitely genera
ted Gorenstein projective cover. We show that every bounded above comp
lex having all terms finitely generated over such a ring has a Gorenst
ein projective cover and we show that these covers are quasi-isomorphi
sms.