Duality pairs, generalized Gorenstein modules, and Ding injective envelopes

Gillespie, James; Iacob, Alina

Comptes Rendus. Mathématique, Tome 360 (2022) no. G4, pp. 381-398.

Let $R$ be a general ring. Duality pairs of $R$-modules were introduced by Holm-Jørgensen. 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 $𝔇=(ℒ,𝒜)$. 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.

Classification : 16D80, 18G25, 18N40

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