  Bivariant (equivariant) K-theory is the standard setting for non-commutative
topology. We may carry over various techniques from homotopy theory and
homological algebra to this setting. Here we do this for some basic notions
from homological algebra: phantom maps, exact chain complexes, projective
resolutions, and derived functors. We introduce these notions and apply them to
examples from bivariant K-theory.
  An important observation of Beligiannis is that we can approximate our
category by an Abelian category in a canonical way, such that our homological
concepts reduce to the corresponding ones in this Abelian category. We compute
this Abelian approximation in several interesting examples, where it turns out
to be very concrete and tractable.
  The derived functors comprise the second tableau of a spectral sequence that,
in favourable cases, converges towards Kasparov groups and other interesting
objects. This mechanism is the common basis for many different spectral
sequences. Here we only discuss the very simple 1-dimensional case, where the
spectral sequences reduce to short exact sequences.
