Gorenstein projective and flat complexes over noetherian rings

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AbstractWe give sufficient conditions on a class of R‐modules \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}$\end{document} in order for the class of complexes of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}$\end{document}‐modules, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$dw \mathcal {C}$\end{document}, to be covering in the category of complexes of R‐modules. More precisely, we prove that if \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}$\end{document} is precovering in R − Mod and if \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}$\end{document} is closed under direct limits, direct products, and extensions, then the class \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$dw \mathcal {C}$\end{document} 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 Cn 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.