Cohomological dimension in pro-$p$-towers

TL;DR

This paper provides a proof, avoiding perfectoid geometry, of Scholze's vanishing theorem for étale cohomology with $ ext{F}_p$-coefficients in a particular pro-$p$-tower, extending beyond the variety's dimension.

Contribution

It offers a new proof of Scholze's vanishing theorem that does not rely on perfectoid geometry, applicable to specific pro-$p$-towers.

Findings

Proof of Scholze's vanishing theorem without perfectoid geometry

Applicable to a specific pro-$p$-tower in characteristic not equal to p

Extends étale cohomology vanishing beyond the dimension of projective varieties

Abstract

We give a proof without use of perfectoid geometry of Scholzes' vanishing theorem of \'etale cohomology with $\mathbb{F}_p$-coefficients beyond the dimension of projective varieties in a specific pro $p$-tower in characteristic not equal to $p$. Final version.