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A motivic integral $p$-adic cohomology

Alberto Merici

TL;DR

This work constructs an integral $p$-adic cohomology theory $RΓ_p$ for smooth varieties over a perfect field of characteristic $p$, framed to factor through Voevodsky's effective motives and to compare with crystalline and rigid cohomology after inverting $p$. The construction uses log-Witt differentials in the Hyodo–Kato style and recovers log de Rham–Witt cohomology under resolution of singularities, while inheriting key motivic properties such as Nisnevich descent, projective bundle formula, and purity; it additionally supports a Künneth formula in the motivic setting. The theory comes with natural maps to crystalline cohomology and, after inverting $p$, to rigid cohomology, and it satisfies a universal descent property for $(\mathbf{A}^1,\mathrm{Nis})$-local objects. Under resolution of singularities, $RΓ_p(X)$ agrees with crystalline cohomology of a compactified pair $(\overline{X},\partial X)$, hence providing a robust bridge between integral $p$-adic cohomology and both crystalline and rigid theories with strong motivic structure.

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 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 Künneth formula for the $p$-adic cohomology of Ertl-Shiho-Sprang.

A motivic integral $p$-adic cohomology

TL;DR

This work constructs an integral -adic cohomology theory for smooth varieties over a perfect field of characteristic , framed to factor through Voevodsky's effective motives and to compare with crystalline and rigid cohomology after inverting . The construction uses log-Witt differentials in the Hyodo–Kato style and recovers log de Rham–Witt cohomology under resolution of singularities, while inheriting key motivic properties such as Nisnevich descent, projective bundle formula, and purity; it additionally supports a Künneth formula in the motivic setting. The theory comes with natural maps to crystalline cohomology and, after inverting , to rigid cohomology, and it satisfies a universal descent property for -local objects. Under resolution of singularities, agrees with crystalline cohomology of a compactified pair , hence providing a robust bridge between integral -adic cohomology and both crystalline and rigid theories with strong motivic structure.

Abstract

We construct an integral -adic cohomology that compares with rigid cohomology after inverting . Our approach is based on the log-Witt differentials of Hyodo-Kato and log-étale motives of Binda-Park-Østvær. In case satisfies resolutions of singularities, we moreover prove that it agrees with the "good" integral -adic cohomology of Ertl-Shiho-Sprang: from this we deduce some interesting motivic properties and a Künneth formula for the -adic cohomology of Ertl-Shiho-Sprang.
Paper Structure (9 sections, 20 theorems, 106 equations)

This paper contains 9 sections, 20 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Recollections
  3. Recollections on log motives
  4. Recollection on reciprocity sheaves
  5. Recollection on log de Rham--Witt sheaves and complex
  6. Complements on box-local sheaves
  7. Application to log de Rham--Witt cohomology
  8. The motivic integral p-adic cohomology
  9. Assuming resolution of singularities

Key Result

Theorem 1.1

There exists an integral $p$-adic cohomology that factors through Voevodsky's stable $\infty$-category of effective motives\begin{tikzcd} \Sm(k)\ar[rr,"R\Gamma_p"]\ar[dr] &&\cD(W(k))^{op}\\ &\DMinf(k,\Z)\ar[ur]. \end{tikzcd}For all $X\in \mathbf{Sm}(k)$, there is a canonical map to crystalli that factors through $R\Gamma_{\rm crys}(\overline{X},\partial X)$ whenever $X$ admits a smooth par

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Theorem 3.3
  • Remark 3.4
  • ...and 35 more