The lower extension groups and quotient categories

TL;DR

This paper explores the relationship between lower extension groups in relative homological algebra and their representations as Hom groups in quotient categories, providing new insights into their algebraic structures.

Contribution

It establishes isomorphisms between lower extension groups and Hom groups in Verdier and dg quotient categories, linking different homological frameworks.

Findings

Lower extension groups are isomorphic to suspended Hom groups in Verdier quotients.

These groups are also isomorphic to negative cohomology groups in dg quotient categories.

The results unify various perspectives in relative homological algebra.

Abstract

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.