Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-27T19:59:57.353Z Has data issue: false hasContentIssue false
  • English
  • Français

The localization theorem for framed motivic spaces

Part of: (Co)homology theory

Published online by Cambridge University Press:  05 February 2021

Marc Hoyois
Marc Hoyois*
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040Regensburg, [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the analog of the Morel–Voevodsky localization theorem for framed motivic spaces. We deduce that framed motivic spectra are equivalent to motivic spectra over arbitrary schemes, and we give a new construction of the motivic cohomology of arbitrary schemes.

Keywords

motivic homotopy theoryframed correspondencesmotivic cohomology

MSC classification

Primary: 14F42: Motivic cohomology; motivic homotopy theory
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 1 , January 2021 , pp. 1 - 11
Check for updates
Copyright
© The Author(s) 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author was partially supported by NSF grant DMS-1761718.

References

Ayoub, J., Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, I, Astérisque 315 (2008).Google Scholar
Ayoub, J., La réalisation étale et les opérations de Grothendieck, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), 1–145.CrossRefGoogle Scholar
Bachmann, T. and Hoyois, M., Norms in motivic homotopy theory, Astérisque, to appear. Preprint (2020), arXiv:1711.03061.Google Scholar
Cisinski, D.-C. and Déglise, F., Triangulated categories of mixed motives, Springer Monographs in Mathematics (Springer, 2019).CrossRefGoogle Scholar
Clausen, D. and Mathew, A., Hyperdescent and étale K-theory, Preprint (2019), arXiv:1905.06611v2.Google Scholar
Déglise, F., Jin, F. and Khan, A. A., Fundamental classes in motivic homotopy theory, Preprint (2018), arXiv:1805.05920.Google Scholar
Elmanto, E., Hoyois, M., Khan, A. A., Sosnilo, V. and Yakerson, M., Motivic infinite loop spaces, Preprint (2019), arXiv:1711.05248v5.Google Scholar
Elmanto, E., Hoyois, M., Khan, A. A., Sosnilo, V. and Yakerson, M., Framed transfers and motivic fundamental classes, J.~Topol. 13 (2020), 460–500.CrossRefGoogle Scholar
Grothendieck, A., Éléments de Géométrie Algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie, Publ. Math. Inst. Hautes Études Sci. 28 (1966).CrossRefGoogle Scholar
Grothendieck, A., Éléments de Géométrie Algébrique: IV. Étude locale des schémas et des morphismes de schémas. Quatrième partie, Publ. Math. Inst. Hautes Études Sci. 32 (1967).Google Scholar
Gruson, L., Une propriété des couples henséliens, Publications mathématiques et informatique de Rennes, no. 4 (1972), Exposé no.~10.Google Scholar
Hoyois, M., A quadratic refinement of the Grothendieck–Lefschetz–Verdier trace formula, Algebr. Geom. Topol. 14 (2014), 3603–3658.CrossRefGoogle Scholar
Hoyois, M., The étale symmetric Künneth theorem, Preprint (2018), arXiv:1810.00351v2.Google Scholar
Levine, M., Techniques of localization in the theory of algebraic cycles, J. Algebraic Geom. 10 (2001), 299–363.Google Scholar
Lurie, J., Higher Algebra, September 2017, http://www.math.harvard.edu/lurie/papers/HA.pdf.Google Scholar
Morel, F. and Voevodsky, V., $\mathbf{A}^1$-homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci. 90 (1999), 45–143.CrossRefGoogle Scholar
Spitzweck, M., A commutative $\mathbf{P}^1$-spectrum representing motivic cohomology over Dedekind domains, Mém. Soc. Math. Fr. 157 (2018).Google Scholar