Duality pairs, generalized Gorenstein modules, and Ding injective envelopes

James Gillespie; Alina Iacob

doi:10.5802/crmath.306

Algèbre

Duality pairs, generalized Gorenstein modules, and Ding injective envelopes

James Gillespie ¹ ; Alina Iacob ²

¹ J.G. Ramapo College of New Jersey School of Theoretical and Applied Science 505 Ramapo Valley Road Mahwah, NJ 07430 U.S.A.
² A.I. Department of Mathematical Sciences Georgia Southern University Statesboro (GA) 30460-8093 U.S.A.

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 381-398.

Résumé

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.

Reçu le : 2021-09-23
Révisé le : 2021-11-21
Accepté le : 2021-11-28
Publié le : 2022-04-26

DOI : 10.5802/crmath.306

Classification : 16D80, 18G25, 18N40

Affiliations des auteurs :

James Gillespie ¹ ; Alina Iacob ²

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {James Gillespie and Alina Iacob},
     title = {Duality pairs, generalized {Gorenstein} modules, and {Ding} injective envelopes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {381--398},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.306},
     language = {en},
}

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James Gillespie; Alina Iacob. Duality pairs, generalized Gorenstein modules, and Ding injective envelopes. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 381-398. doi : 10.5802/crmath.306. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.306/

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