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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Gysin triangle via localization and $A^1$-homotopy invariance

Authors: Gonçalo Tabuada and Michel Van den Bergh
Journal: Trans. Amer. Math. Soc. 370 (2018), 421-446
MSC (2010): Primary 14A22, 14C15, 14F42, 18D20, 19D55
Published electronically: August 15, 2017
MathSciNet review: 3717985
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Abstract: Let $X$ be a smooth scheme, $Z$ a smooth closed subscheme, and $U$ the open complement. Given any localizing and $\mathbb {A}^1$-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 étale descent result concerning noncommutative mixed motives with rational coefficients.

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Additional Information

Gonçalo Tabuada
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 – and – Departamento de Matemática e Centro de Matemática e Aplicações, Faculdade de Ciências e Tecnologia, Universidade Nova de Lsiboa, Quinta da Torre, 2829-516, Caparica, Portugal
MR Author ID: 751291

Michel Van den Bergh
Affiliation: Departement WNI, Universiteit Hasselt, 3590 Diepenbeek, Belgium
MR Author ID: 176980

Keywords: Localization, $A^1$-homotopy, dg category, algebraic $K$-theory, periodic cyclic homology, algebraic spaces, motivic homotopy theory, (noncommutative) mixed motives, Nisnevich and étale descent, relative cellular spaces, noncommutative algebraic geometry
Received by editor(s): November 30, 2015
Received by editor(s) in revised form: January 20, 2016, and April 5, 2016
Published electronically: August 15, 2017
Additional Notes: The first author was partially supported by the NSF CAREER Award #1350472 and by the Portuguese Foundation for Science and Technology grant PEst-OE/MAT/UI0297/2014
The second author is a senior researcher at the Fund for Scientific Research, Flanders
Article copyright: © Copyright 2017 American Mathematical Society