Hostname: page-component-cb9f654ff-rkzlw Total loading time: 0 Render date: 2025-08-24T13:45:26.922Z Has data issue: false hasContentIssue false
  • English
  • Français

THE BOREL CHARACTER

Part of: Forms and linear algebraic groups $K$-theory of forms $K$-theory in geometry

Published online by Cambridge University Press:  26 July 2021

Frédéric Déglise Open the ORCID record for Frédéric Déglise [Opens in a new window]  and
Jean Fasel
Frédéric Déglise
Affiliation:
ENS de Lyon, UMPA, CNRS, 46 allée d’Italie, 69364 Lyon Cedex 07, France, (frederic.deglise@ens-lyon.fr) URL: http://perso.ens-lyon.fr/frederic.deglise/
Jean Fasel
Affiliation:
Institut Fourier - UMR 5582, Université Grenoble-Alpes, CS 40700, 38058 Grenoble Cedex 9, France (Jean.Fasel@univ-grenoble-alpes.fr), URL: https://www-fourier.ujf-grenoble.fr/∼faselj/
Get access Rights & Permissions [Opens in a new window]

Abstract

The main purpose of this article is to define a quadratic analogue of the Chern character, the so-called Borel character, that identifies rational higher Grothendieck-Witt groups with a sum of rational Milnor-Witt (MW)-motivic cohomologies and rational motivic cohomologies. We also discuss the notion of ternary laws due to Walter, a quadratic analogue of formal group laws, and compute what we call the additive ternary laws, associated with MW-motivic cohomology. Finally, we provide an application of the Borel character by showing that the Milnor-Witt K-theory of a field F embeds into suitable higher Grothendieck-Witt groups of F modulo explicit torsion.

Keywords

motivic homotopy theoryHermitian K-theoryMW-motivic cohomologycharacteristic classes

MSC classification

Primary: 19E15: Algebraic cycles and motivic cohomology 19E20: Relations with cohomology theories 19E99: None of the above, but in this section 19G38: Hermitian $K$-theory, relations with $K$-theory of rings 19G99: None of the above, but in this section
Secondary: 11E04: Quadratic forms over general fields 11E39: Bilinear and Hermitian forms 11E70: $K$-theory of quadratic and Hermitian forms 11E81: Algebraic theory of quadratic forms; Witt groups and rings

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 22 , Issue 2 , March 2023 , pp. 747 - 797
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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.)

Article purchase

Temporarily unavailable

References

Ananyevskiy, A., The special linear version of the projective bundle theorem, Compos. Math., 151(3) (2015), 461–501.CrossRefGoogle Scholar
Ananyevskiy, A., Stable operations and cooperations in derived Witt theory with rational coefficients, Ann. $K$ -Theory, 2(4) (2017), 517–560.CrossRefGoogle Scholar
Ananyevskiy, A., $\mathrm{SL}$ -oriented cohomology theories, arXiv:1901.01597, 2019.Google Scholar
Ananyevskiy, A., Levine, M. and Panin, I., Witt sheaves and the $\eta$ -inverted sphere spectrum, J. Topol., 10 (2) (2017), 370–385.CrossRefGoogle Scholar
Asok, A., Doran, B. and Fasel, J., Smooth models of motivic spheres and the clutching construction, Int. Math. Res. Not. IMRN, (6) (2017), 1890–1925.Google Scholar
Asok, A. and Fasel, J., An explicit KO-degree map and applications, J. Topol., 10(1) (2017), 268–300, arXiv:1403.4588.CrossRefGoogle Scholar
Bachmann, T., Calmès, B., Déglise, F., Fasel, J. and Østvær, P. A., Milnor-Witt motives, arXiv:2004.06634, 2020.Google Scholar
Bass, H. and Roy, A., Lectures on Topics in Algebraic $K$ -Theory, Tata Institute of Fundamental Research Lectures on Mathematics, No. 41 (Tata Institute of Fundamental Research, Bombay, 1967). Notes by Amit Roy.Google Scholar
Borel, A., Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4), 7 (1974), 235–272 (1975).CrossRefGoogle Scholar
Cisinski, D.-C., Descente par éclatements en $K$ -théorie invariante par homotopie, Ann. Math. (2), 177(2) (2013), 425–448.CrossRefGoogle Scholar
Cisinski, D.-C. and Déglise, F., Triangulated Categories of Mixed Motives, Springer Monographs in Mathematics (Springer, Cham (Switzerland), 2019), arXiv:0912.2110, version 3.CrossRefGoogle Scholar
Déglise, F., Orientation theory in arithmetic geometry, in $K$ -Theory—Proceedings of the International Colloquium, Mumbai, 2016 (Hindustan Book Agency, New Delhi, 2018), 239–347.Google Scholar
Déglise, F. and Fasel, J., MW-motivic complexes, arXiv:1708.06095, 2017.Google Scholar
Déglise, F. and Fasel, J., The Milnor-Witt motivic ring spectrum and its associated theories, arXiv:1708.06102, 2017.Google Scholar
Déglise, F., Fasel, J., Khan, A. and Jin, F., On the rational motivic homotopy category, Journal de l’École polytechnique — Mathématiques, 8 (2021), 533–583.CrossRefGoogle Scholar
Fulton, W., Intersection Theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Vol. 2 (Springer, Berlin, 1998).CrossRefGoogle Scholar
Gepner, D. and Snaith, V., On the motivic spectra representing algebraic cobordism and algebraic $K$ -theory, Doc. Math., 14 (2009), 359–396.CrossRefGoogle Scholar
Karoubi, Max and Villamayor, Orlando, $K$ -théorie hermitienne , C. R. Acad. Sci. Paris Sér. A-B, 272 (1971), A1237–A1240.Google Scholar
Kolderup, H. A., Remarks on classical number theoretic aspects of Milnor-Witt $K$ -theory, arXiv:1906.07506, 2019.Google Scholar
Levine, M., Motivic Euler characteristics and Witt-valued characteristic classes, Nagoya Math. J., 236 (2019), 1–60.CrossRefGoogle Scholar
Levine, M. and Morel, F., Algebraic Cobordism , Springer Monographs in Mathematics (Springer, Berlin, 2007).Google Scholar
Macdonald, I. G., Symmetric Functions and Hall Polynomials , second ed., Oxford Classic Texts in the Physical Sciences (Oxford University Press, New York, 2015). With contribution by Zelevinsky, A. V. and a foreword by Richard Stanley.Google Scholar
Milnor, J., Algebraic $K$ -theory and quadratic forms, Invent. Math., 9 (1969/70), 318–344.CrossRefGoogle Scholar
Morel, F., On the motivic ${\pi}_0$ of the sphere spectrum, in Axiomatic, Enriched and Motivic Homotopy Theory, NATO Sci. Ser. II Math. Phys. Chem., Vol. 131 (Kluwer Academic, Dordrecht, The Netherlands, 2004), 219–260.CrossRefGoogle Scholar
Morel, F., ${A}^1$ -Algebraic Topology over a Field, Lecture Notes in Mathematics, Vol. 2052 (Springer, Heidelberg, Germany, 2012).CrossRefGoogle Scholar
Panin, I. and Walter, C., Quaternionic Grassmannians and Borel classes in algebraic geometry, arXiv:1011.0649, 2010.Google Scholar
Panin, I. and Walter, C., On the algebraic cobordism spectra MSL and MSp, arXiv: 1011.0651, 2018.Google Scholar
Panin, I. and Walter, C., On the motivic commutative ring spectrum BO, St. Petersbg. Math. J., 30(6) (2019), 933–972.CrossRefGoogle Scholar
Riou, J., Catégorie homotopique stable d’un site suspendu avec intervalle, Bull. Soc. Math. France, 135(4) (2007), 495–547.CrossRefGoogle Scholar
Riou, J., Algebraic K-theory, ${\mathbf{A}}^1$ -homotopy and Riemann-Roch theorems, J. Topol., 3(2) (2010), 229–264.CrossRefGoogle Scholar
Röndigs, O. and Østvær, P. A., Slices of Hermitian $K$ -theory and Milnor’s conjecture on quadratic forms, Geom. Topol., 20(2) (2016), 1157–1212.CrossRefGoogle Scholar
Scharlau, W., Quadratic and Hermitian Forms, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 270 (Springer, Berlin, 1985).CrossRefGoogle Scholar
Schlichting, M., Hermitian $K$ -theory, derived equivalences and Karoubi’s fundamental theorem, J. Pure Appl. Algebra, 221(7) (2017), 1729–1844.CrossRefGoogle Scholar
Schlichting, M. and Tripathi, G. S., Geometric models for higher Grothendieck-Witt groups in ${A}^1$ -homotopy theory, Math. Ann., 362(3–4) (2015), 1143–1167.CrossRefGoogle Scholar
Soulé, C., Opérations en $K$ -théorie algébrique, Can. J. Math., 37(3) (1985), 488–550.CrossRefGoogle Scholar
Spitzweck, M., A commutative ${\mathbb{P}}^1$ -spectrum representing motivic cohomology over Dedekind domains, Mém. Soc. Math. Fr. (N.S.), (157) (2018), 1–110.Google Scholar
Suslin, A. A., Homology of ${\mathrm{GL}}_n$ , characteristic classes and Milnor $K$ -theory, in Algebraic $K$ -Theory, Number Theory, Geometry and Analysis, Lecture Notes in Math., Vol. 1046 (Springer, Berlin, 1984), 357–375.Google Scholar
Vishik, A., Stable and unstable operations in algebraic cobordism, Ann. Sci. Éc. Norm. Supér. (4), 52(3) (2019), 561–630.CrossRefGoogle Scholar
Voevodsky, V., ${\mathbf{A}}^1$ -homotopy theory, in Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 579–604.Google Scholar
Wall, C. T. C., Surgery of non-simply-connected manifolds, Ann. Math. (2), 84 (1966), 217–276.CrossRefGoogle Scholar