Gorenstein projective and flat complexes over noetherian rings

We give sufficient conditions on a class of R-modules C in order for the class of complexes of C-modules, dwC, to be covering in the category of complexes of R-modules. More precisely, we prove that if C is precovering in R - Mod and if C is closed under direct limits, direct products, and extensions, then the class dwC is covering in Ch(R). Our first application concerns the class of Gorenstein flat modules. We show that when the ring R is two sided noetherian, a complex C is Gorenstein flat if and only if each module C-n is Gorenstein flat. If moreover every direct product of Gorenstein flat modules is a Gorenstein flat module, then the class of Gorenstein flat complexes is covering. We consider Gorenstein projective complexes as well. We prove that if R is a commutative noetherian ring of finite Krull dimension, then the class of Gorenstein projective complexes coincides with that of complexes of Gorenstein projective modules. We also show that if R is commutative noetherian with a dualizing complex then every right bounded complex has a Gorenstein projective precover. (C) 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim