Gorenstein projective and flat complexes over noetherian rings

Abstract

Key words Precover, cover, Gorenstein flat complex, Gorenstein projective complex MSC (2010) 18G35, 18G25We 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 − M od 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 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.

On Gorenstein Flat Preenvelopes of Complexes

“…We show that over commutative and Noetherian rings with a dualizing complex every complex admits a Gorenstein flat preenvelope. This complements the work of Enochs, Estrada and Iacob who proved in [8] that over such rings every complex admits a Gorenstein flat (pre)cover. The method we use to show that the class of Gorenstein flat complexes is preenveloping in the category of complexes works in a more general setting, and so the existence of preenvelopes by many important classes of complexes is obtained.…”

Gorenstein complexes and recollements from cotorsion pairs

“…For example, see [GR99]. Enochs, Estrada, and Iacob have shown this for Gorenstein projective complexes over a commutative Noetherian ring admitting a dualizing complex [EE05]. Moreover, we see from [YL11] that this is true for any ring R!…”

Kaplansky Classes and Cotorsion Theories of Complexes

“…Gorenstein injective) R-modules, then T is the class of Gorenstein projective (resp. Gorenstein injective) complexes in R , see [7,29]. Some of the above classes has already been studied in literature with different notations.…”