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Please use this identifier to cite or link to this item: http://hdl.handle.net/1942/26283

Title: The Gysin triangle via localization and A(1)-homotopy invariance
Authors: Tabuada, Goncalo
Van den Bergh, Michel
Issue Date: 2018
Publisher: AMER MATHEMATICAL SOC
Citation: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 370(1), p. 421-446
Abstract: Let X be a smooth scheme, Z a smooth closed subscheme, and U the open complement. Given any localizing and A(1)-homotopy invariant of dg categories E, we construct an associated Gysin triangle relating the value of E at the dg categories of perfect complexes of X, Z, and U. In the particular case where E is homotopy K-theory, this Gysin triangle yields a new proof of Quillen's localization theorem, which avoids the use of devissage. As a first application, we prove that the value of E at a smooth scheme belongs to the smallest (thick) triangulated subcategory generated by the values of E at the smooth projective schemes. As a second application, we compute the additive invariants of relative cellular spaces in terms of the bases of the corresponding cells. Finally, as a third application, we construct explicit bridges relating motivic homotopy theory and mixed motives on the one side with noncommutative mixed motives on the other side. This leads to a comparison between different motivic Gysin triangles as well as to an etale descent result concerning noncommutative mixed motives with rational coefficients.
Notes: [Tabuada, Goncalo] MIT, Dept Math, Cambridge, MA 02139 USA. [Tabuada, Goncalo] Univ Nova Lsiboa, Fac Ciencias & Tecnol, Dept Matemat, P-2829516 Quinta Da Torre, Caparica, Portugal. [Tabuada, Goncalo] Univ Nova Lsiboa, Fac Ciencias & Tecnol, Ctr Matemat & Aplicacoes, P-2829516 Quinta Da Torre, Caparica, Portugal. [Van den Bergh, Michel] Univ Hasselt, Dept WNI, B-3590 Diepenbeek, Belgium.
URI: http://hdl.handle.net/1942/26283
DOI: 10.1090/tran/6956
ISI #: 000414149300014
ISSN: 0002-9947
Category: A1
Type: Journal Contribution
Appears in Collections: Research publications

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