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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
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 > 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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| Date of first compliant deposit: | 16 September 2016 | ||||||||
| 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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