Motivic p-adic tame cohomology

TL;DR

This paper develops a comparison between tame motives and log-étale motives over fields of positive characteristic, leading to the construction of a spectrum representing mod p^m tame motivic cohomology, confirming conjectures by Hübner and Schmidt.

Contribution

It introduces a new comparison functor between tame and log-étale motives and constructs an $E_$-ring spectrum for mod p^m tame motivic cohomology, extending previous results.

Findings

Constructed a comparison functor between tame and log-étale motives.

Built an $E_$-ring spectrum representing mod p^m tame motivic cohomology.

Proved results on tame motivic cohomology confirming conjectures.

Abstract

We construct a comparison functor between ($\mathbf{A}^1$-local) tame motives and ($\overline{\square}$-local) log-\'etale motives over a field $k$ of positive characteristic. This generalizes Binda--Park--{\O}stv{\ae}r's comparison for the Nisnevich topology. As a consequence, we construct an $E_\infty$-ring spectrum $H\mathbb{Z}/p^m$ representing mod $p^m$ tame motivic cohomology: the existence of this ring spectrum and the usual properties of motives imply some results on tame motivic cohomology, which were conjectured by H\"ubner--Schmidt.