Gorenstein projective and flat complexes over noetherian rings |
| |
Authors: | E Enochs S Estrada A Iacob |
| |
Institution: | 1. Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA;2. Departamento de Matemática Aplicada, Universidad de Murcia, Campus del Espinardo, Espinardo (Murcia) 30100, Spain;3. Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460, USA |
| |
Abstract: | We give sufficient conditions on a class of R‐modules $\mathcal {C}We give sufficient conditions on a class of R‐modules $\mathcal {C}$ in order for the class of complexes of $\mathcal {C}$‐modules, $dw \mathcal {C}$, to be covering in the category of complexes of R‐modules. More precisely, we prove that if $\mathcal {C}$ is precovering in R ? Mod and if $\mathcal {C}$ is closed under direct limits, direct products, and extensions, then the class $dw \mathcal {C}$ 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. |
| |
Keywords: | Precover cover Gorenstein flat complex Gorenstein projective complex MSC (2010) 18G35 18G25 |
|
|