2015/10/15 by Gonçalo Tabuada, Goncalo Tabuada, Michel Van den Bergh +2
Mathematics · #14A22 #14C15 #14F42 #18D20 #19D55 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.AG #math.AT #math.KT #math.RA #math.RT #msc:14A22 #msc:14C15 #msc:14F42 #msc:18D20 #msc:19D55
paper · pdf · doi:10.48550/arxiv.1510.04677
25 pages. Revised version
openalex publication_date 2015/10/15 · arxiv created 2015/10/25 · arxiv updated 2015/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.