Completing perfect complexes
Abstract
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 nonaffine 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2018
 arXiv:
 arXiv:1805.10751
 Bibcode:
 2018arXiv180510751B
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Category Theory;
 18E30 (primary);
 14F05;
 16E35;
 55P42 (secondary)
 EPrint:
 35 pages, with appendices by Tobias Barthel, Henning Krause and Bernhard Keller. Version 3: Added a new appendix by Tobias Barthel and Henning Krause that refines for a separated noetherian scheme the results of the main text. Added a discussion of completions of abelian categories and their derived categories. Version 4: Minor revision, correcting some examples