The Gysin triangle via localization and A[superscript 1]-homotopy invariance
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S0002-9947-2017-06956-7.pdf
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Author(s)
Van den Bergh, Michel
Trigo Neri Tabuada, Goncalo Jorge
Date Issued
January 2018
Journal
Transactions of the American Mathematical Society
Publisher
American Mathematical Society (AMS)
Citation
Tabuada, Gonçalo, and Michel Van den Bergh. “The Gysin Triangle via Localization and A[superscript 1]-Homotopy Invariance.” Transactions of the American Mathematical Society, vol. 370, no. 1, Aug. 2017, pp. 421–46. © American Mathematical Society
Version
Final published version
Abstract
Let X be a smooth scheme, Z a smooth closed subscheme, and
U the open complement. Given any localizing and A[superscript 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1090/TRAN/6956
