Gluing approximable triangulated categories

Abstract

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 $\mathsf {D}_{\mathsf {qc}}( X )$ is approximable when X is a quasi compact, separated scheme 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, in conjunction with the groundwork laid in [16], has interesting applications in existing and forthcoming work by the authors.