De Rham Comparison and Poincaré Duality for Rigid Varieties

Abstract

Over any smooth algebraic variety over a p-adic local field k, we construct the de Rham comparison isomorphisms for the étale cohomology with partial compact support of de Rham \({\mathbb {Z}}_p\)-local systems, and show that they are compatible with Poincaré duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over k. In particular, we prove finiteness of étale cohomology with partial compact support of any \({\mathbb {Z}}_p\)-local systems, and establish the Poincaré duality for such cohomology after inverting p.