Duality pairs, generalized Gorenstein modules, and Ding injective envelopes

TL;DR

This paper develops a generalized Gorenstein homological algebra framework based on semi-complete duality pairs, leading to new results on Ding injective modules and cotorsion pairs over arbitrary rings.

Contribution

It introduces a relative Gorenstein theory for semi-complete duality pairs, extending existing concepts and establishing the completeness of Ding injective modules as cotorsion pairs.

Findings

Ding injective modules form the right side of a complete cotorsion pair over any ring.

The theory generalizes the homological framework of AC-Gorenstein modules.

Completeness of the Gorenstein flat cotorsion pair is established over any ring.

Abstract

Let $R$ be a general ring. Duality pairs of $R$-modules were introduced by Holm-Jorgensen. Most examples satisfy further properties making them what we call semi-complete duality pairs in this paper. We attach a relative theory of Gorenstein homological algebra to any given semi-complete duality pair $\mathfrak{D} = (\mathcal{L},\mathcal{A})$. This generalizes the homological theory of the AC-Gorenstein modules defined by Bravo-Gillespie-Hovey, and we apply this to other semi-complete duality pairs. The main application is that the Ding injective modules are the right side of a complete (perfect) cotorsion pair, over any ring. Completeness of the Gorenstein flat cotorsion pair over any ring arises from the same duality pair.