Relative Homological Algebra via Truncations

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 $\mathcal{A}$ and fix a class of “injective objects” $\mathcal{I}$. 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 $D(\mathcal{A};\mathcal{I})$, we need more: the split error term must vanish. This is the case when $\mathcal{I}$ 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${}^{\ast }$-$n$ axiom for abelian categories. Various concrete examples such as Gorenstein homological algebra and purity are also discussed.