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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 | Source: | 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. | Keywords: | Localization; A(1)-homotopy; dg category; algebraic K-theory; periodic cyclic homology; algebraic spaces; motivic homotopy theory; (noncommutative) mixed motives; Nisnevich and etale descent; relative cellular spaces; noncommutative algebraic geometry;localization; A(1)-homotopy; dg category; algebraic K-theory; periodic cyclic homology; algebraic spaces; motivic homotopy theory; (noncommutative) mixed motives; Nisnevich and etale descent; relative cellular spaces; noncommutative algebraic geometry | Document URI: | http://hdl.handle.net/1942/26283 | ISSN: | 0002-9947 | e-ISSN: | 1088-6850 | DOI: | 10.1090/tran/6956 | ISI #: | 000414149300014 | Rights: | (C) 2017 American Mathematical society | Category: | A1 | Type: | Journal Contribution | Validations: | ecoom 2018 |
Appears in Collections: | Research publications |
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1510.04677.pdf | Non Peer-reviewed author version | 406.81 kB | Adobe PDF | View/Open |
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