2024/11/11 by Guido Bosco, Bosco, Guido
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2411.07355
openalex publication_date 2024/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let C be a complete algebraically closed extension of ℚp, and let \mathfrakX be a smooth formal scheme over OC. By the work of Bhatt--Morrow--Scholze, it is known that when \mathfrakX is proper, the length of the torsion in the integral p-adic étale cohomology of the generic fiber \mathfrakXC 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 \mathfrakX is affine, the torsion in the integral p-adic étale cohomology of \mathfrakXC can even be expressed as a functor of the special fiber, unlike in the proper case. As a consequence, we show that, surprisingly, if \mathfrakX is affine, the integral p-adic étale cohomology groups of \mathfrakXC 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.