# Relative homological algebra via truncations

## TL;DR

This paper provides a homotopy theoretical interpretation of Spaltenstein's truncation method for resolutions in unbounded chain complexes, extending it to a relative setting within abelian categories.

## Contribution

It introduces a Quillen pair framework capturing the truncation and resolution process, enabling a homotopy-theoretic approach to relative homological algebra.

## Key findings

- The truncation process can be modeled by a Quillen pair of adjoint functors.
- The construction yields a relative derived category D(A; I) under certain conditions.
- The split error term vanishes when I is the class of all injective modules, but not in general.

## Abstract

To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spaltenstein solved this problem for chain complexes of R-modules by truncating further and further to the left, resolving the pieces, and gluing back the partial resolutions. Our aim is to give a homotopy theoretical interpretation of this procedure, which may be extended to a relative setting. We work in an arbitrary abelian category A and fix a class I of "injective objects". We show that Spaltenstein's construction can be captured by a pair of adjoint functors between unbounded chain complexes and towers of non-positively graded ones. This pair of adjoint functors forms what we call a Quillen pair and the above process of truncations, partial resolutions, and gluing, gives a meaningful way to resolve complexes in a relative setting up to a split error term. In order to do homotopy theory, and in particular to construct a well behaved relative derived category D(A; I), we need more: the split error term must vanish. This is the case when I is the class of all injective R-modules but not in general, not even for certain classes of injectives modules over a Noetherian ring. The key property is a relative analogue of Roos's AB4*-n axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed.