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THE BOREL CHARACTER

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

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

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