Torsion in $p$-adic étale cohomology: remarks and conjectures

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 étale 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 étale 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 étale 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 étale cohomology of a broader class of rigid-analytic varieties over $C$.