Uniform local constancy of \'etale cohomology of rigid analytic varieties

TL;DR

This paper establishes uniform local constancy of étale cohomology for rigid analytic varieties, demonstrating that nearby neighborhoods share the same cohomological properties across different primes, extending previous results by Huber.

Contribution

It provides a uniform refinement of Orgogozo's theorem, proving the existence of small neighborhoods with consistent étale cohomology across all primes different from the residue characteristic.

Findings

Proved $ ext{l}$-independence of local étale cohomology for rigid analytic varieties.

Showed existence of neighborhoods with identical étale cohomology for subschemes and their neighborhoods.

Extended Huber's result by establishing uniformity across all relevant primes.

Abstract

We prove some $\ell$-independence results on local constancy of \'etale cohomology of rigid analytic varieties. As a result, we show that a closed subscheme of a proper scheme over an algebraically closed complete non-archimedean field has a small open neighborhood in the analytic topology such that, for every prime number $\ell$ different from the residue characteristic, the closed subscheme and the open neighborhood have the same \'etale cohomology with $\mathbb Z/\ell \mathbb Z$-coefficients. The existence of such an open neighborhood for each $\ell$ was proved by Huber. A key ingredient in the proof is a uniform refinement of a theorem of Orgogozo on the compatibility of the nearby cycles over general bases with base change.