Completing perfect complexes
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Mathematische Zeitschrift
Completing perfect complexes With appendices by Tobias Barthel and Bernhard Keller Henning Krause1 Dedicated to the memory of Ragnar–Olaf Buchweitz. Received: 23 March 2019 / Accepted: 14 January 2020 © The Author(s) 2020
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. Keywords Completion · Cauchy sequence · Derived category · Triangulated category · Morphic enhancement · Perfect complex · Coherent ring · Noetherian scheme · Ring spectrum Mathematics Subject Classification Primary 18E30; Secondary 14F05 · 16E35 · 55P42
1 Introduction This note proposes a new method to complete a triangulated category, and we apply this to categories of perfect complexes [17]. For any category C, we introduce its sequential
Tobias Barthel was supported by the the Danish National Research Foundation (DNRF92) and the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska–Curie grant agreement No. 751794.
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Henning Krause [email protected] Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany
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H. Krause
completion C, which is a categorical analogue of the construction of the real numbers from the rationals via equivalence classes of Cauchy sequences, following Cantor and Méray [5,27]. When a ring is right coherent, then the category mod of finitely presented modules is abelian and one can consider its bounded derived category Db (mod ), which contains the category of perfect complexes Dper () as a full triangulated subcategory. The following theorem describes Db (mod ) as a completion of Dper (). Theorem 1.1 For a right coherent ring there is a canonical triangle equivalence b
per () −→ Db (mod ) D ∼
which sends a Cauchy sequence in Dper () to its colimit. The description of Db (mod ) as a completion extends to non-affine schemes. Thus for a noetherian scheme X there is a canonical triangle equivalence b
per (X) −∼ D → Db (coh X).
In particular, this provides a direct construction of the singularity category (in the sen
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