Article contents
Motivic Steenrod operations in characteristic p
Published online by Cambridge University Press: 13 November 2020
Abstract
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.
- Type
- Topology
- Information
- Creative Commons
- 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
References
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20201112123122521-0672:S2050509420000341:S2050509420000341_inline9.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20201112123122521-0672:S2050509420000341:S2050509420000341_inline10.png?pub-status=live)
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20201112123122521-0672:S2050509420000341:S2050509420000341_inline11.png?pub-status=live)
- 8
- Cited by