Gorenstein flat modules with respect to duality pairs

Abstract

Let X be a class of left R-modules, Y be a class of right R-modules. In this paper, we introduce and study Gorenstein (X , Y)-flat modules as a common generalization of some known modules such as Gorenstein flat modules [9], Gorenstein n-flat modules [22], Gorenstein B-flat modules [7], Gorenstein AC-flat modules [2], Ω-Gorenstein flat modules [10] and so on. We show that the class of all Gorenstein (X , Y)-flat modules have a strong stability. In particular, when (X , Y) is a perfect (symmetric) duality pair, we give some functorial descriptions of Gorenstein (X , Y)-flat dimension, and construct a hereditary abelian model structure on R-Mod whose cofibrant objects are exactly the Gorenstein (X , Y)-flat modules. These results unify the corresponding results of the aforementioned modules.

Model structures and recollements induced by duality pairs

“…In [18], authors introduced and studied a kind of Gorenstein (X , Y)-flat module with respect to two classes of modules X and Y. An R-module M is called Gorenstein (X , Y)-flat if there exists a Y⊗ R −-exact exact sequence…”

“…This fact builds the bridge between duality pairs and cotorsion pairs. In [18,Proposition 2.18], authors gave some equivalent characterizations of GF (X ,Y) (R) under some conditions, where (X , Y) is a complete duality pair. We introduce the definition of Gorenstein X -objects with respect to a cotorsion pair and use G(X ) to represent the class of Gorenstein X -objects in [19].…”

“…We introduce the definition of Gorenstein X -objects with respect to a cotorsion pair and use G(X ) to represent the class of Gorenstein X -objects in [19]. We observe that GF (X ,Y) (R) is exactly G(X ) under the conditions of [18,Proposition 2.18]. So we will obtain exact model structures associated to GF (X ,Y) (R).…”

“…The notion of a duality pair of R-modules was introduced by Holm and Jφrgensen in [15]. Duality pairs exist extensively (refer to [13,15,18]). Indeed, duality pairs are related to purity and the existence of covers and envelopes, which makes duality pairs be very useful in relative homological algebra and attract wide attention.…”

“…As a result, we have tried to find Frobenius pairs by Gorenstein objects with respect to cotorsion pairs in [5], and obtained some meaningful conclusions. Wang and coauthors introduced and studied Gorenstein flat modules with respect to duality pairs, which enriches Gorenstein homological algebra with respect to a duality pair in [18]. However, it is well known that a perfect duality pair can induce a perfect cotorsion pair.…”