Motivic Steenrod operations at the characteristic via infinite ramification

TL;DR

The paper develops mod-$p$ motivic Steenrod operations in characteristic $p$ by leveraging a motivic version of Nizioł's purity theorem and Levine's purity, facilitated by deep ramification. It constructs natural bistable operators $P^n_{ij}$ and $B^n_{ij}$ on mod-$p$ motivic cohomology with Adem and Cartan relations, and shows these operate compatibly with motivic and syntomic frameworks, extending beyond the Chow diagonal. A key technical advance is a motivic Nizioł-type purity result over the infinitely ramified base $\mathcal{O}=\mathbb{Z}_p[p^{1/p^\infty}]$, enabling a canonical map from rational to mod-$p$ endomorphisms and a spectrum-level Cartan formula. The paper then derives geometric applications in positive characteristic, including non-smoothable cycles, obstructions to lifting to $\mathrm{MGL}$, counterexamples to the integral Tate conjecture at the characteristic, and a Wu formula for motivic and syntomic cohomology, highlighting deep connections between arithmetic ramification, motivic homotopy theory, and cycle-theoretic phenomena in characteristic $p$.

Abstract

We construct motivic power operations on the mod-$p$ motivic cohomology of $\Fb_p$-schemes using a motivic refinement of Nizioł'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.

Figures (1)

• Figure 1: Illustration of the class $\{t\} \in \mathrm{CH}^1(\mathbb{G}_m, 1) = H^1_{\mathbb{A}^1}(\mathbb{G}_m, \mathbb{Z}(1))$. The cycles $[A]$ and $[B]$ are the graphs of the two functions $\lambda \to \lambda$ and $\lambda \to \lambda^{-1}$ from $\mathbb{G}_m$ to $\mathbb{A}^1$, respectively. The two cycles are elements of $z^1(\mathbb{G}_m,1)$ because each of them meets the two faces, represented by the horizontal lines at $0$ and $1$, transversely. The difference $[B]-[A]$ is a cycle in the cycle complex, and therefore it gives rise to a class $\{t\} := [B]-[A] \in \mathrm{CH}^1(\mathbb{G}_m, 1)$.

Theorems & Definitions (117)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3: Power operations and norms
• Remark 1.4: An element called $\tau$
• Remark 1.5: Other fields of characteristic $p$
• Theorem 1.7
• Conjecture 1.8
• Conjecture 1.9: The Hopkins--Morel isomorphism aka the motivic Quillen theorem
• Example 2.1: Algebraic and Hermitian $K$-theory
• Example 2.2: $\ell$-adic cohomology