  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.
