Motivic Steenrod operations at the characteristic via infinite ramification

TL;DR

This paper constructs motivic Steenrod operations at characteristic p using a motivic refinement of Nizio{ }l's theorem, providing new tools and results in motivic cohomology with applications to algebraic cycles and Tate conjectures.

Contribution

It introduces motivic power operations on mod-p motivic cohomology over schemes in characteristic p, extending previous work to all bidegrees with key geometric applications.

Findings

Constructed motivic Steenrod operations satisfying naturality, Adem relations, and Cartan formula.

Provided examples of non-(quasi-)smoothable algebraic cycles at characteristic p.

Addressed the motivic Steenrod problem and offered a counterexample to the integral crystalline Tate conjecture.

Abstract

We construct motivic power operations on the mod-$p$ motivic cohomology of $\Fb_p$-schemes using a motivic refinement of Nizio{\l}'s theorem. The key input is a purity theorem for motivic cohomology established by Levine. Our operations satisfy the expected properties (naturality, Adem relations, and the Cartan formula) for all bidegrees, generalizing previous results of Primozic which were only know along the ``Chow diagonal.'' We offer geometric applications of our construction: 1) an example of non-(quasi-)smoothable algebraic cycle at the characteristic, 2) an answer to the motivic Steenrod problem at the characteristic, 3) a counterexample to the integral version of a crystalline Tate conjecture.