Motivic p-adic tame cohomology

Abstract

We construct a comparison functor between (${\mathbf{A}}^1$-local) tame motives and ($\overline{\square }$-local) log-étale motives over a field $k$ of positive characteristic. This generalizes Binda--Park--Østvær'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 Hübner--Schmidt.

Minor changes. Accepted for publication in Bull. Lond. Math. Soc