  Given a bounded-above cochain complex of modules over a ring, it is standard
to replace it by a projective resolution, and it is classical that doing so can
be very useful. Recently, a modified version of this was introduced in
triangulated categories other than the derived category of a ring. A
triangulated category is approximable if this modified procedure is possible.
Not surprisingly this has proved a powerful tool. For example: the fact that
the derived category of a quasi compact, separated scheme is approximable has
led to major improvements on old theorems due to Bondal, Van den Bergh and
Rouquier.
  In this article we prove that, under weak hypotheses, the recollement of two
approximable triangulated categories is approximable. In particular, this shows
many of the triangulated categories that arise in noncommutative algebraic
geometry are approximable. Furthermore, the lemmas and techniques developed in
this article form a powerful toolbox which has interesting applications in
existing and forthcoming work by the authors.
