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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

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Official URL: https://doi.org/10.1016/j.jpaa.2016.12.026

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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
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:
DateEvent
July 2017Published
21 December 2016Available
24 August 2016Accepted
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
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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