Uniform local constancy of étale cohomology of rigid analytic varieties

Abstract

We prove some $\ell$-independence results on local constancy of étale 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 étale 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.

55 pages