Duality for Arithmetic $p$-adic Pro-\'etale Cohomology of Analytic Spaces

TL;DR

This paper proves a duality property for the arithmetic $p$-adic pro-étale cohomology of smooth partially proper spaces over a $p$-adic field, confirming a conjecture and connecting geometric duality with Galois descent.

Contribution

It establishes a duality for $p$-adic pro-étale cohomology of certain spaces, deriving it from geometric duality on the Fargues-Fontaine curve using Galois descent.

Findings

Proves duality for arithmetic $p$-adic pro-étale cohomology of smooth spaces.

Confirms a conjecture by Colmez, Gilles, and Nizioł.

Connects geometric duality with Galois descent techniques.

Abstract

Let $K$ be a finite extension of $\mathbb{Q}_p$. We prove that the arithmetic $p$-adic pro-\'etale cohomology of smooth partially proper spaces over $K$ satisfies a duality, as conjectured by Colmez, Gilles and Nizio{\l}. We derive it from the geometric duality on the Fargues-Fontaine curve by Galois descent techniques of Fontaine.