The Gysin triangle via localization and A1-homotopy invariance

TL;DR

This paper develops a Gysin triangle framework for localizing and A1-homotopy invariant dg categories, providing new proofs, computations, and bridges between motivic and noncommutative motives.

Contribution

It introduces a Gysin triangle construction for dg categories, offers a new proof of Quillen's localization theorem, and connects motivic and noncommutative motives.

Findings

Provides a new proof of Quillen's localization theorem.

Computes invariants of relative cellular spaces.

Establishes bridges between motivic and noncommutative motives.

Abstract

Let X be a smooth scheme, Z a smooth closed subscheme, and U the open complement. Given any localizing and A1-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.