2020/04/13 by Henning Krause · 1 citation
Mathematics · #Abelian category #Abelian group #Advanced Topics in Algebra #Affine transformation #Algebra over a field #Algebraic structures and combinatorial models #Biproduct #Bounded function #Category of groups #Closed category #Concrete category #Derived category #Discrete mathematics #Finitely-generated abelian group #Functor #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Noetherian #Noetherian ring #Pure mathematics #Ring (chemistry) #Scheme (mathematics) #Triangulated category
paper · pdf · doi:10.1007/s00209-020-02490-z
openalex publication_date 2020/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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.