TY - JOUR

T1 - Hyperdescent and étale K-theory

AU - Clausen, Dustin

AU - Mathew, Akhil

N1 - 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.

PY - 2021/9

Y1 - 2021/9

N2 - 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.

AB - 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.

U2 - 10.1007/s00222-021-01043-3

DO - 10.1007/s00222-021-01043-3

M3 - Journal article

AN - SCOPUS:85104140421

VL - 225

SP - 981

EP - 1076

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

SN - 0020-9910

IS - 3

ER -