Effective motives with and without transfers in characteristic $p$

TL;DR

This paper establishes equivalences between different categories of effective motives in rigid analytic and algebraic geometry over fields of characteristic p, clarifying their relationships with and without transfers.

Contribution

It proves the equivalence of categories of effective motives with and without transfers in characteristic p, including stable and relative versions, unifying various motivic frameworks.

Findings

Equivalence between effective motives with and without transfers in characteristic p

In characteristic zero, equivalence of rigid analytic motives with and without transfers

Establishment of stable and relative versions of the main equivalence

Abstract

We prove the equivalence between the category $\mathbf{RigDM}_{et}^{eff}(K,\mathbb{Q})$ of effective motives of rigid analytic varieties over a perfect complete non-archimedean field $K$ and the category $\mathbf{RigDM}_{Frobet}^{eff}(K,\mathbb{Q})$ which is obtained by localizing the category of motives without transfers $\mathbf{RigDA}_{et}^{eff}(K,\mathbb{Q})$ over purely inseparable maps. In particular, we obtain an equivalence between $\mathbf{RigDM}_{et}^{eff}(K,\mathbb{Q})$ and $\mathbf{RigDA}_{et}^{eff}(K,\mathbb{Q})$ in the characteristic $0$ case and an equivalence between $\mathbf{DM}_{et}^{eff}(K,\mathbb{Q})$ and $\mathbf{DA}_{Frobet}^{eff}(K,\mathbb{Q})$ of motives of algebraic varieties over a perfect field $K$. We also show a relative and a stable version of the main statement.