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Motivic Steenrod operations in characteristic p

Part of: (Co)homology theory $K$-theory in geometry Homotopy theory

Published online by Cambridge University Press:  13 November 2020

Eric Primozic
Eric Primozic*
Affiliation:
Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada; E-mail: primozic@ualberta.ca

Abstract

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For a prime p and a field k of characteristic $p,$ we define Steenrod operations $P^{n}_{k}$ on motivic cohomology with $\mathbb {F}_{p}$ -coefficients of smooth varieties defined over the base field $k.$ We show that $P^{n}_{k}$ is the pth power on $H^{2n,n}(-,\mathbb {F}_{p}) \cong CH^{n}(-)/p$ and prove an instability result for the operations. Restricted to mod p Chow groups, we show that the operations satisfy the expected Adem relations and Cartan formula. Using these new operations, we remove previous restrictions on the characteristic of the base field for Rost’s degree formula. Over a base field of characteristic $2,$ we obtain new results on quadratic forms.

MSC classification

Primary: 14F42: Motivic cohomology; motivic homotopy theory 19E15: Algebraic cycles and motivic cohomology
Secondary: 55P43: Spectra with additional structure ($E_infty$, $A_infty$, ring spectra, etc.)
Type
Topology
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e52
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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