## Abstract

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

Original language | English |
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Pages (from-to) | 421-446 |

Number of pages | 26 |

Journal | Transactions of the American Mathematical Society |

Volume | 370 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2018 |

## Keywords

- (noncommutative) mixed motives
- A -homotopy
- Algebraic K-theory
- Algebraic spaces
- Dg category
- Localization
- Motivic homotopy theory
- Nisnevich and étale descent
- Noncommutative algebraic geometry
- Periodic cyclic homology
- Relative cellular spaces

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