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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
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: |
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| 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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