The Gysin triangle via localization and A[superscript 1]-homotopy invariance
Author(s)
Van den Bergh, Michel; Trigo Neri Tabuada, Goncalo Jorge
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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.
Date issued
2018-01Department
Massachusetts Institute of Technology. Department of MathematicsJournal
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
ISSN
0002-9947
1088-6850