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The homotopy fixed point theorem and the Quillen–Lichtenbaum conjecture in Hermitian K-theory

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Berrick, A. J., Karoubi, M., Schlichting, Marco and Østvær, P. A. (2015) The homotopy fixed point theorem and the Quillen–Lichtenbaum conjecture in Hermitian K-theory. Advances in Mathematics, 278 . pp. 34-55. doi:10.1016/j.aim.2015.01.018 ISSN 0001-8708.

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Official URL: http://dx.doi.org/10.1016/j.aim.2015.01.018

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Abstract

Let X be a noetherian scheme of finite Krull dimension, having 2 invertible in its ring of regular functions, an ample family of line bundles, and a global bound on the virtual mod-2 cohomological dimensions of its residue fields. We prove that the comparison map from the hermitian K-theory of X to the homotopy fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence in general, and an integral equivalence when X has no formally real residue field. We also show that the comparison map between the higher Grothendieck-Witt (hermitian K-) theory of X and its ´etale version is an isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum conjecture in K-theory. Applications compute higher Grothendieck-Witt groups of complex algebraic varieties and rings of 2-integers in number fields, and hence values of Dedekind zeta-functions.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Homotopy theory, K-theory, Grothendieck groups, Krull rings
Journal or Publication Title: Advances in Mathematics
Publisher: Academic Press
ISSN: 0001-8708
Official Date: 25 June 2015
Dates:
DateEvent
25 June 2015Published
28 January 2015Accepted
Volume: 278
Page Range: pp. 34-55
DOI: 10.1016/j.aim.2015.01.018
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 5 September 2017
Date of first compliant Open Access: 5 September 2017
Funder: National Science Foundation (U.S.) (NSF)
Grant number: 0906290

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