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AB-contexts and stability for Gorenstein flat modules with respect to semidualizing modules. (English) Zbl 1317.13029

Summary: 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.

MSC:

13C60 Module categories and commutative rings
13D02 Syzygies, resolutions, complexes and commutative rings
13C05 Structure, classification theorems for modules and ideals in commutative rings
13C11 Injective and flat modules and ideals in commutative rings
13D05 Homological dimension and commutative rings
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