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-étale motives of Binda–Park–Østvær. In case $k$ satisfies resolution 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 Künneth formula for the $p$-adic cohomology of Ertl–Shiho–Sprang.

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