For a certain full additive subcategory 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 cohomological 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.

中文翻译：

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较低的扩展组和商类别

对于加性类别 A 的某个完全加性子类别 X，定义了相对同调代数中的较低扩展群。我们证明这些群与 X 的有界同伦范畴的 Verdier 商范畴中的悬置 Hom 群同构。或者，这些群与 dg 商范畴 A 中 Hom 配合物的负上同调群同构/X，其中 A 和 X 都被视为集中在零度的 dg 类别。