AB-Contexts and Stability for Gorenstein Flat Modules with Respect to Semidualizing Modules

TL;DR

This paper studies G_C-flat modules over a commutative noetherian ring with semidualizing modules, establishing their properties within weak AB-contexts and deriving approximation results and module construction procedures.

Contribution

It introduces the concept of G_C-flat modules within weak AB-contexts and proves approximation and module construction results related to semidualizing modules.

Findings

G_C-flat modules form part of a weak AB-context.

Existence of Auslander-Buchweitz approximations for modules of finite G_C-flat dimension.

Procedures for building modules from complete resolutions are validated.

Abstract

We investigate the properties of categories of G_C-flat R-modules where C is a semidualizing module over a commutative noetherian ring R. We prove that the category of all G_C-flat R-modules is part of a weak AB-context, in the terminology of Hashimoto. In particular, this allows us to deduce the existence of certain Auslander-Buchweitz approximations for R-modules of finite G_C-flat dimension. We also prove that two procedures for building R-modules from complete resolutions by certain subcategories of G_C-flat R-modules yield only the modules in the original subcategories.