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Hermitian K-theory, derived equivalences and Karoubi's fundamental theorem
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Schlichting, Marco (2017) Hermitian K-theory, derived equivalences and Karoubi's fundamental theorem. Journal of Pure and Applied Algebra, 221 (7). pp. 1729-1844. doi:10.1016/j.jpaa.2016.12.026 ISSN 0022-4049.
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Official URL: https://doi.org/10.1016/j.jpaa.2016.12.026
Abstract
Within the framework of dg categories with weak equivalences and duality that have uniquely 2-divisible mapping complexes, we show that higher Grothendieck-Witt groups (aka. hermitian K-groups) are invariant under derived equivalences and that Morita exact sequences induce long exact sequences of Grothendieck-Witt groups. This implies an algebraic Bott sequence and a new proof and generalisation of Karoubi’s Fundamental Theorem. For the higher Grothendieck-Witt groups of vector bundles of (possibly singular) schemes X with an ample family of line-bundles such that 1 2 ∈ Γ(X, OX), we obtain Mayer-Vietoris long exact sequences for Nisnevich coverings and blowups along regularly embedded centers, projective bundle formulas, and a Bass fundamental theorem. For coherent Grothendieck-Witt groups, we obtain a localization theorem analogous to Quillen’s K′ -localization theorem.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | k-groups, Grothendieck groups | ||||||||
Journal or Publication Title: | Journal of Pure and Applied Algebra | ||||||||
Publisher: | Elsevier Science BV | ||||||||
ISSN: | 0022-4049 | ||||||||
Official Date: | July 2017 | ||||||||
Dates: |
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Volume: | 221 | ||||||||
Number: | 7 | ||||||||
Page Range: | pp. 1729-1844 | ||||||||
DOI: | 10.1016/j.jpaa.2016.12.026 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 16 September 2016 | ||||||||
Date of first compliant Open Access: | 21 December 2017 | ||||||||
Funder: | Institut des hautes études scientifiques (Paris, France), Max-Planck-Gesellschaft zur Förderung der Wissenschaften [Max Planck Society for the Advancement of Science], National Science Foundation (U.S.) (NSF), Engineering and Physical Sciences Research Council (EPSRC) | ||||||||
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