On Gorenstein Flat Preenvelopes of Complexes

Abstract

In this paper we show that if the class f of R-modules is closed under well ordered direct limits, then the class f is preenveloping in the category of Rmodules if and only if the class dwf is preenveloping in the category of Rcomplexes, where dwf denotes the class of all complexes with all components in f. As an immediate consequence, we get that over commutative and Noetherian rings with dualizing complexes every complex admits a Gorenstein flat preenvelope. MATHEMATICS SUBJECT CLASSIFICATION (2010). 16E05, 18G35. KEYWORDS. (pre)envelopes, Gorenstein injective modules (complexes), FP-injective modules, Gorenstein flat modules (complexes).

Gorenstein flat and projective (pre)covers

“…By Proposition 2, this class is also closed under direct limits, and therefore it is a covering class in Ch(R). By [14] Prop. 3.2, this is the class of Gorenstein flat complexes.…”

“…It is known ( [14]) that over a right coherent ring R, a complex G is Gorenstein flat if and only if each module G n is Gorenstein flat.…”

“…We can prove now the existence of special Gorenstein projective precovers over a right coherent and left perfect ring R. We recall that (by [2], Proposition 3.7), if R is right coherent and any flat R-module has finite projective dimension, then any Gorenstein projective module is also Gorenstein flat. Then by [13] and by [14], over a right coherent ring that is also left n-perfect, every Gorenstein projective complex is also a Gorenstein flat complex. We use the notation GorP roj for the class of Gorenstein projective complexes and we denote by GorF lat the class of Gorenstein flat complexes.…”