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Completing perfect complexes
- Source :
- Mathematische Zeitschrift. 296:1387-1427
- Publication Year :
- 2020
- Publisher :
- Springer Science and Business Media LLC, 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.
- 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
Subjects
Details
- ISSN :
- 14321823 and 00255874
- Volume :
- 296
- Database :
- OpenAIRE
- Journal :
- Mathematische Zeitschrift
- Accession number :
- edsair.doi.dedup.....cf5ad8ce360b1d99a87b3a14db0b4cc7