The gysin triangle via localization and A 1 -homotopy invariance

Gonçalo Tabuada, Michel Van Den Bergh

Research output: Contribution to journalArticle

4 Citations (Scopus)

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 languageEnglish
Pages (from-to)421-446
Number of pages26
JournalTransactions of the American Mathematical Society
Volume370
Issue number1
DOIs
Publication statusPublished - 1 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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