Relative Derived Category with respect to a Subcategory

TL;DR

This paper introduces the concept of relative derived categories with respect to a subcategory, extending existing theorems and providing new interpretations and applications in homological algebra.

Contribution

It extends Gao and Zhang's theorem to the bounded below case and interprets relative derived functors as morphisms within these categories.

Findings

Established a triangle-equivalence extending previous results.

Interpreted relative derived functors as morphisms in the new categories.

Provided two applications demonstrating the utility of the framework.

Abstract

The notion of relative derived category with respect to a subcategory is introduced. A triangle-equivalence, which extends a theorem of Gao and Zhang [Gorenstein derived categories, \emph{J. Algebra} \textbf{323} (2010) 2041-2057] to the bounded below case, is obtained. Moreover, we interpret the relative derived functor $\mathrm{Ext}_{\mathcal{X}\mathcal {A}}(-,-)$ as the morphisms in such derived category and give two applications.