Compactly supported $p$-adic pro-\'etale cohomology of analytic varieties

TL;DR

This paper investigates the properties of compactly supported $p$-adic pro-étale cohomology for smooth rigid analytic varieties, establishing a comparison with syntomic cohomology and deriving a fundamental diagram.

Contribution

It introduces a comparison theorem between compactly supported $p$-adic pro-étale and syntomic cohomologies, enhancing understanding of their relationship in rigid analytic geometry.

Findings

Proves a comparison theorem in a stable range

Establishes a fundamental diagram relating different cohomologies

Links pro-étale cohomology with syntomic, Hyodo-Kato, and ${ m B}^+_{ m dr}$-cohomologies

Abstract

We study properties of compactly supported $p$-adic pro-\'etale cohomology of smooth partially proper rigid analytic varieties. In particular, we prove a comparison theorem, in a stable range, with compactly supported syntomic cohomology, which is built from compactly supported Hyodo-Kato and ${\mathcal B}^+_{\rm dr}$-cohomologies. We derive from that a (limited version of a) fundamental diagram.