Algèbre

Duality pairs, generalized Gorenstein modules, and Ding injective envelopes

Gillespie, James¹  ; Iacob, Alina²

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, Tome 360 (2022) no. G4, 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.

Détail

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

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

Bibliographie

[1] Bravo, Daniel; Gillespie, James; Hovey, Mark The stable module category of a general ring (2014) (https://arxiv.org/abs/1405.5768)

[2] Bravo, Daniel; Gillespie, James; Pérez, Marco A. Locally type ${FP}_n$ and $n$-coherent categories (2019) (https://arxiv.org/abs/1908.10987)

[3] Bravo, Daniel; Pérez, Marco A. Finiteness conditions and cotorsion pairs, J. Pure Appl. Algebra, Volume 221 (2017) no. 6, pp. 1249-1267 | MR | DOI | Zbl

[4] Cortés-Izurdiaga, Manuel; Šaroch, Jan Module classes induced by complexes and $\lambda$-pure-injective modules (2021) (https://arxiv.org/abs/2104.08602)

[5] Enochs, Edgar; Iacob, Alina; Jenda, Overtoun M. G. Closure under Transfinite Extensions, Ill. J. Math., Volume 51 (2007) no. 2, pp. 561-569 | Zbl | MR

[6] Enochs, Edgar; Jenda, Overtoun M. G. Relative homological algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter, 2000 | DOI | Zbl

[7] Estrada, Sergio; Gillespie, James The projective stable category of a coherent scheme, Proc. R. Soc. Edinb., Sect. A, Math., Volume 149 (2019) no. 1, pp. 15-43 | Zbl | MR | DOI

[8] Estrada, Sergio; Iacob, Alina; Pérez, Marco A. Model structures and relative Gorenstein flat modules and chain complexes, Categorical, homological and combinatorial methods in algebra (Contemporary Mathematics), Volume 751, American Mathematical Society, 2020, pp. 135-175 | Zbl | MR | DOI

[9] García Rozas, Juan R. Covers and envelopes in the category of complexes of modules, CRC Research Notes in Mathematics, 407, Chapman & Hall/CRC, 1999

[10] Gillespie, James Model structures on modules over Ding-Chen rings, Homology Homotopy Appl., Volume 12 (2010) no. 1, pp. 61-73 | Zbl | MR | DOI

[11] Gillespie, James How to construct a Hovey triple from two cotorsion pairs, Fundam. Math., Volume 230 (2015) no. 3, pp. 281-289 | Zbl | MR | DOI

[12] Gillespie, James Gorenstein complexes and recollements from cotorsion pairs, Adv. Math., Volume 291 (2016), pp. 859-911 | Zbl | MR | DOI

[13] Gillespie, James Hereditary abelian model categories, Bull. Lond. Math. Soc., Volume 48 (2016) no. 6, pp. 895-922 | Zbl | MR | DOI

[14] Gillespie, James Models for homotopy categories of injectives and Gorenstein injectives, Commun. Algebra, Volume 45 (2017) no. 6, pp. 2520-2545 | Zbl | MR | DOI

[15] Gillespie, James On Ding injective, Ding projective and Ding flat modules and complexes, Rocky Mt. J. Math., Volume 47 (2017) no. 8, pp. 2641-2673 | Zbl | MR

[16] Gillespie, James Gorenstein AC-projective complexes, J. Homotopy Relat. Struct., Volume 13 (2018) no. 4, pp. 769-791 | Zbl | MR | DOI

[17] Gillespie, James Duality pairs and stable module categories, J. Pure Appl. Algebra, Volume 223 (2019) no. 8, pp. 3425-3435 | Zbl | MR | DOI

[18] Göbel, Rüdiger; Trlifaj, Jan Approximations and Endomorphism Algebras of Modules, De Gruyter Expositions in Mathematics, 41, Walter de Gruyter, 2006 | DOI

[19] Holm, Henrik Gorenstein homological dimensions, J. Pure Appl. Algebra, Volume 189 (2004), pp. 167-193 | Zbl | MR | DOI

[20] Holm, Henrik; Jørgensen, Peter Cotorsion pairs induced by duality pairs, J. Commut. Algebra, Volume 1 (2009) no. 4, pp. 621-633 | MR | Zbl

[21] Hovey, Mark Cotorsion pairs, model category structures, and representation theory, Math. Z., Volume 241 (2002), pp. 553-592 | DOI | Zbl | MR

[22] Iacob, Alina Generalized Gorenstein modules (to appear in Algebra Colloq.)

[23] Mao, Lixin; Ding, Nanqing Gorenstein FP-injective and Gorenstein flat modules, J. Algebra Appl., Volume 7 (2008) no. 4, pp. 491-506 | Zbl | MR

[24] Neeman, Amnon Triangulated categories, Annals of Mathematics Studies, 148, Princeton University Press, 2001 | DOI

[25] Neeman, Amnon The homotopy category of flat modules, and Grothendieck duality, Invent. Math., Volume 174 (2008) no. 2, pp. 255-308 | Zbl | MR | DOI

[26] Prest, Mike Purity, spectra and localisation, Encyclopedia of Mathematics and Its Applications, 121, Cambridge University Press, 2009 | DOI

[27] Rosický, Jiří Generalized Brown representability in homotopy categories, Theory Appl. Categ., Volume 14 (2005) no. 19, pp. 451-479 | Zbl | MR

[28] Šaroch, Jan; Šťovíček, Jan Singular compactness and definability for $\Sigma$-cotorsion and Gorenstein modules, Sel. Math., New Ser., Volume 26 (2020) no. 2, 23, 40 pages | Zbl | MR

[29] Šťovíček, Jan Deconstructibility and the Hill lemma in Grothendieck categories, Forum Math., Volume 25 (2013) no. 1, pp. 193-219 | Zbl | MR

[30] Šťovíček, Jan On purity and applications to coderived and singularity categories (2014) (https://arxiv.org/abs/1412.1615)

[31] Yang, Gang; Liu, Zhongkui; Liang, Li Ding projective and Ding injective modules, Algebra Colloq., Volume 20 (2013) no. 4, pp. 601-612 | Zbl | MR | DOI