The Gysin triangle via localization and 𝐴¹-homotopy invariance

Tabuada, Gonçalo; Van den Bergh, Michel

doi:10.1090/tran/6956

The Gysin triangle via localization and $A^1$-homotopy invariance
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by Gonçalo Tabuada and Michel Van den Bergh PDF

Trans. Amer. Math. Soc. 370 (2018), 421-446 Request permission

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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