  This note proposes a new method to complete a triangulated category, which is
based on the notion of a Cauchy sequence. We apply this to categories of
perfect complexes. It is shown that the bounded derived category of finitely
presented modules over a right coherent ring is the completion of the category
of perfect complexes. The result extends to non-affine noetherian schemes and
gives rise to a direct construction of the singularity category. The parallel
theory of completion for abelian categories is compatible with the completion
of derived categories. There are three appendices. The first one by Tobias
Barthel discusses the completion of perfect complexes for ring spectra. The
second one by Tobias Barthel and Henning Krause refines for a separated
noetherian scheme the description of the bounded derived category of coherent
sheaves as a completion. The final appendix by Bernhard Keller introduces the
concept of a morphic enhancement for triangulated categories and provides a
proper foundation for completing a triangulated category.
