When are the classes of Gorenstein modules (co)tilting?

Junpeng Wang; Zhongkui Liu; Renyu Zhao

doi:10.5802/crmath.639

Article de recherche - Algèbre

When are the classes of Gorenstein modules (co)tilting?
[Quand les classes de modules de Gorenstein sont-elles (co)basculantes ?]

Junpeng Wang ¹ ; Zhongkui Liu ¹ ; Renyu Zhao ¹

¹ Department of Mathematics, Northwest Normal University, Lanzhou 730070, People’s Republic of China

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1301-1325.

Résumé
Abstract

Pour la classe des modules Gorenstein-projectifs (respectivement $G$ -injectifs et $G$ -plats), nous étudions et réglons les questions de savoir quand la seconde est basculante et les autres cobasculantes. Les applications vont dans trois directions. La première est d’obtenir la coïncidence entre les propriétés de ces classes d’être, respectivement, 1-basculante et bousculante, ainsi que la propriété d’être 1-cobasculante et cobousculante. La deuxième consiste à caractériser les modules de Gorenstein via des modules finiment engendrés, ce qui prouve que les anneaux noethériens à gauche de dimension globale de Gorenstein gauche finie satisfont la première conjecture de la dimension finitiste et un résultat lié à une question posée par Bazzoni dans [J. Algebra 320 (2008) 4281-4299]. Le dernier objectif est de donner de nouvelles caractérisations des domaines de Dedekind et de Prüfer et des algèbres d’Artin de Gorenstein commutatives, ainsi que des anneaux de Gorenstein généraux (éventuellement non commutatifs) et des anneaux de Ding–Chen.

For the class of Gorenstein projective (resp. injective and flat) modules, we investigate and settle the questions when the middle class is tilting and the other ones are cotilting. The applications have in three directions. The first is to obtain the coincidence between the 1-tilting and silting property, as well as the 1-cotilting and cosilting property of such classes respectively. The second is to characterize Gorenstein modules via finitely generated modules, which provides a proof of that left Noetherian rings with finite left Gorenstein global dimension satisfy First Finitistic Dimension Conjecture and a result related to a question posed by Bazzoni in [J. Algebra 320 (2008) 4281-4299]. The last is to give some new characterizations of Dedekind and Prüfer domains and commutative Gorenstein Artin algebras as well as general (possibly not commutative) Gorenstein rings and Ding–Chen rings.

Reçu le : 2023-04-22
Révisé le : 2023-09-07
Accepté le : 2024-04-15
Publié le : 2024-11-14

DOI : 10.5802/crmath.639

Classification : 18G25, 18E45, 16E60, 16E65
Keywords: Gorenstein projective (resp. injective and flat) module, (co)tilting class, finitistic dimension conjecture, (strongly) finite type
Mot clés : Module Gorenstein-projectif (respectivement $G$-injectif et $G$-plat), classe de (co)basculement, conjecture de la dimension finitiste, type (fortement) fini

Affiliations des auteurs :

Junpeng Wang ¹ ; Zhongkui Liu ¹ ; Renyu Zhao ¹

¹ Department of Mathematics, Northwest Normal University, Lanzhou 730070, People’s Republic of China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Junpeng Wang and Zhongkui Liu and Renyu Zhao},
     title = {When are the classes of {Gorenstein} modules (co)tilting?},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1301--1325},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.639},
     language = {en},
}

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Junpeng Wang; Zhongkui Liu; Renyu Zhao. When are the classes of Gorenstein modules (co)tilting?. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1301-1325. doi : 10.5802/crmath.639. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.639/

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