Arithmetic duality for $p$-adic pro-\'etale cohomology of analytic curves

TL;DR

This paper establishes a Poincaré duality for the arithmetic p-adic pro-étale cohomology of smooth dagger curves over p-adic fields, linking geometric comparison theorems with classical dualities.

Contribution

It proves a novel Poincaré duality for p-adic pro-étale cohomology of analytic curves, including compatibility checks and functional analytic properties.

Findings

Poincaré duality for p-adic pro-étale cohomology of curves

Cohomology is nuclear Fréchet and compactly supported cohomology is of compact type

Compatibility of dualities verified via reduction to the ghost circle

Abstract

We prove a Poincar\'e duality for arithmetic $p$-adic pro-\'etale cohomology of smooth dagger curves over finite extensions of ${\mathbf Q}_p$. We deduce it, via the Hochschild-Serre spectral sequence, from geometric comparison theorems combined with Tate and Serre dualities. The compatibility of all the products involved is checked via reduction to the ghost circle, for which we also prove a Poincar\'e duality (showing that it behaves like a proper smooth analytic variety of dimension $1/2$). Along the way we study functional analytic properties of arithmetic $p$-adic pro-\'etale cohomology and prove that the usual cohomology is nuclear Fr\'echet and the compactly supported one -- of compact type.