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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
DOI: https://doi.org/10.1090/tran/6956
Published electronically: August 15, 2017
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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
Email: tabuada@math.mit.edu

Michel Van den Bergh
Affiliation: Departement WNI, Universiteit Hasselt, 3590 Diepenbeek, Belgium
Email: michel.vandenbergh@uhasselt.be

DOI: https://doi.org/10.1090/tran/6956
Keywords: Localization, $A^1$-homotopy, dg category, algebraic $K$-theory, periodic cyclic homology, algebraic spaces, motivic homotopy theory, (noncommutative) mixed motives, Nisnevich and {\'e}tale 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

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