Torsion in $p$-adic \'etale cohomology: remarks and conjectures

TL;DR

This paper investigates torsion phenomena in the integral p-adic étale cohomology of affine formal schemes over p-adic fields, revealing finiteness properties and proposing conjectures for broader classes.

Contribution

It extends known bounds from proper to affine cases, showing torsion can be expressed via the special fiber and suggesting new conjectures for rigid-analytic varieties.

Findings

Torsion in affine case can be expressed as a functor of the special fiber.

Integral p-adic étale cohomology groups of affine schemes have finite torsion subgroups.

Proposes conjectures for torsion behavior in broader classes of rigid-analytic varieties.

Abstract

Let $C$ be a complete algebraically closed extension of $\mathbb{Q}_p$, and let $\mathfrak{X}$ be a smooth formal scheme over $\mathcal{O}_C$. By the work of Bhatt--Morrow--Scholze, it is known that when $\mathfrak{X}$ is proper, the length of the torsion in the integral $p$-adic \'etale cohomology of the generic fiber $\mathfrak{X}_C$ is bounded above by the length of the torsion in the crystalline cohomology of its special fiber. In this note, we focus on the non-proper case and observe that when $\mathfrak{X}$ is affine, the torsion in the integral $p$-adic \'etale cohomology of $\mathfrak{X}_C$ can even be expressed as a functor of the special fiber, unlike in the proper case. As a consequence, we show that, surprisingly, if $\mathfrak{X}$ is affine, the integral $p$-adic \'etale cohomology groups of $\mathfrak{X}_C$ have finite torsion subgroups. We discuss further applications and propose conjectures predicting the torsion in the integral $p$-adic \'etale cohomology of a broader class of rigid-analytic varieties over $C$.