1. Completing perfect complexes
- Author
-
Krause, Henning
- Subjects
Noetherian ,Pure mathematics ,Triangulated category ,General Mathematics ,Perfect complex ,01 natural sciences ,Completion ,Derived category ,010305 fluids & plasmas ,Coherent sheaf ,Coherent ring ,Mathematics::Category Theory ,0103 physical sciences ,0101 mathematics ,Abelian group ,Morphic enhancement ,Mathematics ,Ring (mathematics) ,Cauchy sequence ,010102 general mathematics ,Noetherian scheme ,Ring spectrum - 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 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 foundation for completing a triangulated category.
- Published
- 2020