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.

Abstract (VO)
Résumé

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.

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.

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

@article{CRMATH_2019__357_11-12_832_0,
     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},
}

TY  - JOUR
AU  - Xiaofa Chen
AU  - Xiao-Wu Chen
TI  - The lower extension groups and quotient categories
JO  - Comptes Rendus. Mathématique
PY  - 2019
SP  - 832
EP  - 840
VL  - 357
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2019.11.006
LA  - en
ID  - CRMATH_2019__357_11-12_832_0
ER  -

%0 Journal Article
%A Xiaofa Chen
%A Xiao-Wu Chen
%T The lower extension groups and quotient categories
%J Comptes Rendus. Mathématique
%D 2019
%P 832-840
%V 357
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2019.11.006
%G en
%F CRMATH_2019__357_11-12_832_0

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

[1] A.A. Beilinson Coherent sheaves on $P^{n}$ and problems of linear algebra, Funct. Anal. Appl., Volume 12 (1978) no. 3, pp. 214-216

[2] A.A. Beilinson; J. Bernstein; P. Deligne Faisceaux pervers, Astérisque, vol. 100, Soc. Math. France, Paris, 1982

[3] A. Beligiannis The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and (co)stabilization, Commun. Algebra, Volume 28 (2000), pp. 4547-4596

[4] X.W. Chen Homotopy equivalences induced by balanced pairs, J. Algebra, Volume 324 (2010), pp. 2718-2731

[5] X. Chen; X.W. Chen An informal introduction to dg categories, 2019 | arXiv

[6] V. Drinfeld DG quotients of DG categories, J. Algebra, Volume 272 (2004), pp. 643-691

[7] E.E. Enochs; O.M.G. Jenda Gorenstein injective and projective modules, Math. Z., Volume 220 (1995), pp. 611-633

[8] M. Kalck; D. Yang Relative singularity categories II: dg models, 2018 | arXiv

[9] B. Keller Deriving DG categories, Ann. Sci. Éc. Norm. Supér., Volume 27 (1994) no. 4, pp. 63-102

[10] B. Keller On the cyclic homology of exact categories, J. Pure Appl. Algebra, Volume 136 (1999) no. 1, pp. 1-56

[11] B. Keller; D. Vossieck Sous les catégories dérivées, C. R. Acad. Sci. Paris, Ser. I, Volume 305 (1987), pp. 225-228

[12] J. Rickard Morita theory for derived categories, J. Lond. Math. Soc. (2), Volume 39 (1989), pp. 436-456

Cité par Sources :

Commentaires - Politique