On the torsion in the cohomology of the integral structure sheaf of affinoid adic spaces

TL;DR

The paper proves that for a normal affinoid adic space $X$ over a zero-characteristic non-archimedean field, the cohomology of the integral structure sheaf $H^q(X,\mathcal{O}_X^+)$ is uniformly torsion for all $q\ge1$, i.e. $\mathfrak{m}^N H^q(X,\mathcal{O}_X^+)=0$ for some $N$. The approach reduces the problem to the rigid-analytic ball $\mathbb{B}^d$ via Noether normalization and then uses a rigid-analytic Riemann’s Hebbarkeitssatz together with base-change tools (notably De Jong–Van der Put) to transfer torsion information from the base to $X$. A key technical step is establishing a relation between $\psi_*\mathcal{O}_X^+$ and $\mathcal{O}_{\mathbb{B}^d}^+$ after a suitable power of a local equation, together with a local-cohomology analysis to propagate torsion across fibers. This completes a program initiated by Bartenwerfer and addressed by Hansen–Kedlaya, extending uniform torsion results from smooth to normal affinoid spaces and contributing to the theory of plus-sheafy and diamantine Huber pairs.

Abstract

We prove that the cohomology of the integral structure sheaf of a normal affinoid adic space over a non-archimedean field of characteristic zero is uniformly torsion. This result originated from a remark of Bartenwerfer around the 1980s and it partially answers a recent question of Hansen and Kedlaya (see also Problems 27 and 39 in the Non-Archimedean Scottish Book).

Theorems & Definitions (12)

• Theorem 1.1: Bartenwerfer, Luktebohmert, Hansen-Kedlaya
• Theorem 1.2
• Theorem 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• Proposition 2.4
• Lemma 2.5
• proof
• Claim 2.6