A new proof of a vanishing result due to Berthelot, Esnault, and R\"ulling

TL;DR

This paper provides a more concise proof of a vanishing cohomology result for schemes over mixed characteristic discrete valuation rings, connecting generic and special fibres via Witt vector cohomology.

Contribution

It offers a simplified proof of a known vanishing theorem using modern techniques from Beilinson and Nekovár–Nizioł related to the $h$-topos.

Findings

Simplified proof of the vanishing result.

Clarifies the relationship between generic and special fibres.

Utilizes advanced $h$-topos techniques.

Abstract

The goal of this small note is to give a more concise proof of a result due to Berthelot, Esnault, and R\"ulling. For a regular, proper, and flat scheme $X$ over a discrete valuation ring of mixed characteristic $(0,p)$, it relates the vanishing of the cohomology of the structure sheaf of the generic fibre of $X$ with the vanishing of the Witt vector cohomology of its special fibre. We use as a critical ingredient results and constructions by Beilinson and Nekov\'a\v{r}--Nizio{\l} related to the $h$-topos over a $p$-adic field.