Comptes Rendus
Algebra/Homological algebra
The lower extension groups and quotient categories
Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 832-840.

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.

Published online:
DOI: 10.1016/j.crma.2019.11.006

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
     author = {Xiaofa Chen and Xiao-Wu Chen},
     title = {The lower extension groups and quotient categories},
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     pages = {832--840},
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     year = {2019},
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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.

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