A motivic integral $p$-adic cohomology

TL;DR

This paper develops a new integral p-adic cohomology theory that aligns with rigid cohomology after inverting p, leveraging log-Witt differentials and motives, and establishes its compatibility with existing theories and a Künneth formula.

Contribution

It introduces an integral p-adic cohomology that compares with rigid cohomology and proves its agreement with existing integral p-adic cohomology under certain conditions.

Findings

Constructed a new integral p-adic cohomology theory.

Proved the theory agrees with existing cohomology when resolutions of singularities exist.

Established a K"unneth formula for the p-adic cohomology.

Abstract

We construct an integral $p$-adic cohomology that compares with rigid cohomology after inverting $p$. Our approach is based on the log-Witt differentials of Hyodo-Kato and log-\'etale motives of Binda-Park-{\O}stv{\ae}r. In case $k$ satisfies resolutions of singularities, we moreover prove that it agrees with the "good" integral $p$-adic cohomology of Ertl-Shiho-Sprang: from this we deduce some interesting motivic properties and a K\"unneth formula for the $p$-adic cohomology of Ertl-Shiho-Sprang.