\'Etale cohomology of algebraizable rigid analytic varieties via nearby cycles over general bases

TL;DR

This paper establishes finiteness and comparison theorems for étale cohomology of algebraizable rigid analytic varieties, linking higher direct images to nearby cycles and addressing cases with higher-dimensional targets.

Contribution

It extends finiteness and comparison results in étale cohomology to algebraizable rigid analytic varieties using modifications and nearby cycles over general bases.

Findings

Proves a finiteness theorem for étale cohomology of algebraizable rigid varieties.

Establishes a comparison theorem relating direct images to nearby cycles.

Shows that certain higher direct images are preserved under modifications in the algebraizable case.

Abstract

We prove a finiteness theorem and a comparison theorem in the theory of \'etale cohomology of rigid analytic varieties. By a result of Huber, for a quasi-compact separated morphism of rigid analytic varieties with target being of dimension $\le1$, the compactly supported higher direct image preserves quasi-constructibility. Though the analogous statement for morphisms with higher dimensional target fails in general, we prove that, in the algebraizable case, it holds after replacing the target with a modification. We deduce it from a known finiteness result in the theory of nearby cycles over general bases and a new comparison result, which gives an identification of the compactly supported higher direct image sheaves, up to modification of the target, in terms of nearby cycles over general bases.