The lower extension groups and quotient categories

Xiaofa Chen; Xiao-Wu Chen

doi:10.1016/j.crma.2019.11.006

Algebra/Homological algebra

The lower extension groups and quotient categories
[Groupes d'extension inférieurs et catégories quotient]

Présenté par : Michele Vergne

Xiaofa Chen ¹ ; Xiao-Wu Chen ¹

¹ Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, PR China

Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 832-840.

Résumé
Abstract

Pour certaines sous-catégories pleines additives $X$ d'une catégorie additive $A$ , on définit les groupes d'extension inférieurs en algèbre homologique relative. Nous montrons que ces groupes sont isomorphes aux groupes Hom suspendus dans la catégorie quotient de Verdier de la catégorie homotopique bornée de $A$ par celle de $X$ . Alternativement, ces groupes sont isomorphes aux groupes de cohomologie négatifs des complexes Hom dans le dg-quotient $A / X$ , où $A$ et $X$ sont considérés comme dg-catégories concentrées en degré zéro.

For certain full additive subcategories $X$ of an additive category $A$ , one defines the lower extension groups in relative homological algebra. We show that these groups are isomorphic to the suspended Hom groups in the Verdier quotient category of the bounded homotopy category of $A$ by that of $X$ . Alternatively, these groups are isomorphic to the negative cohomology groups of the Hom complexes in the dg quotient category $A / X$ , where both $A$ and $X$ are viewed as dg categories concentrated in degree zero.

Reçu le : 2019-10-12
Accepté le : 2019-11-12
Publié le : 2019-11-01

DOI : 10.1016/j.crma.2019.11.006

Affiliations des auteurs :

Xiaofa Chen ¹ ; Xiao-Wu Chen ¹

¹ Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, Anhui, PR China

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     author = {Xiaofa Chen and Xiao-Wu Chen},
     title = {The lower extension groups and quotient categories},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {832--840},
     publisher = {Elsevier},
     volume = {357},
     number = {11-12},
     year = {2019},
     doi = {10.1016/j.crma.2019.11.006},
     language = {en},
}

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Xiaofa Chen; Xiao-Wu Chen. The lower extension groups and quotient categories. Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 832-840. doi : 10.1016/j.crma.2019.11.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.11.006/

Bibliographie
Cité par

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