Compactly supported $p$-adic pro-étale cohomology of analytic varieties

TL;DR

The paper defines compactly supported $p$-adic pro-étale cohomology for smooth partially proper rigid analytic varieties and proves a stable-range comparison with compactly supported syntomic cohomology via period morphisms, yielding a fundamental diagram and trace maps. It develops rigid and overconvergent versions of de Rham and Hyodo–Kato cohomologies with compact support, establishing dualities and comparison results with Huber and syntomic theories. The work extends to overconvergent (dagger) varieties, proves local-global compatibility, and provides explicit computations for affine spaces, tori, and Stein spaces, illustrating the theory and enabling geometric period morphisms. It also introduces geometric syntomic cohomology with compact support, relates it to HK and ${f B}^+_{ m dR}$-cohomology, and geometrizes the period morphism within the TVS framework, with applications to trace maps and Poincaré duality in $p$-adic pro-étale cohomology.

Abstract

We study properties of compactly supported $p$-adic pro-étale 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.

Theorems & Definitions (178)

• Theorem 1.2
• Corollary 1.6
• Remark 1.8
• Remark 2.2
• Lemma 2.3
• Remark 2.5
• proof
• Remark 2.7
• Theorem 2.8
• Definition 2.9