Hyperdescent and étale K-theory

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We study the étale sheafification of algebraic K-theory, called étale K-theory. Our main results show that étale K-theory is very close to a noncommutative invariant called Selmer K-theory, which is defined at the level of categories. Consequently, we show that étale K-theory has surprisingly well-behaved properties, integrally and without finiteness assumptions. A key theoretical ingredient is the distinction, which we investigate in detail, between sheaves and hypersheaves of spectra on étale sites.

Original languageEnglish
JournalInventiones Mathematicae
Volume225
Issue number3
Pages (from-to)981-1076
Number of pages96
ISSN0020-9910
DOIs
Publication statusPublished - Sep 2021

Bibliographical note

Funding Information:
We would like to thank Bhargav Bhatt, Lars Hesselholt, and Jacob Lurie for helpful conversations related to this project. We also thank Matthew Morrow, Niko Naumann, and Justin Noel for the related collaborations [12 , 13 ] and many consequent discussions. We are grateful to Adriano C?rdova, Elden Elmanto, and the anonymous referees for many helpful comments on this paper. This work was done while the second author was a Clay Research Fellow, and is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the second author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester. The second author also thanks the University of Copenhagen for its hospitality during multiple visits.

Funding Information:
We would like to thank Bhargav Bhatt, Lars Hesselholt, and Jacob Lurie for helpful conversations related to this project. We also thank Matthew Morrow, Niko Naumann, and Justin Noel for the related collaborations [, ] and many consequent discussions. We are grateful to Adriano Córdova, Elden Elmanto, and the anonymous referees for many helpful comments on this paper. This work was done while the second author was a Clay Research Fellow, and is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the second author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester. The second author also thanks the University of Copenhagen for its hospitality during multiple visits.

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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