Rational analytic syntomic cohomology

TL;DR

This paper introduces rational analytic syntomic cohomology for rigid-analytic varieties over bp, establishing duality, Chern classes, and links to de Rham bundles, extending classical p-adic Hodge theory results.

Contribution

It defines a new cohomology theory for rigid-analytic varieties, connecting syntomic, de Rham, and p-adic Hodge structures with novel duality and Chern class theories.

Findings

Established Poincare9 duality for the new cohomology

Identified vector bundles with de Rham bundles on the Fargues--Fontaine curve

Recovered classical comparison theorems in p-adic Hodge theory

Abstract

We define and study the rational analytic syntomification $X^{\mathrm{Syn}}$ of a partially proper rigid-analytic variety $X$ over $\mathbb{Q}_p$. We establish Poincar\'e duality and a theory of first Chern classes for the resulting cohomology theory, identify vector bundles on $X^{\mathrm{Syn}}$ with de Rham bundles on the Fargues--Fontaine curve of $X^{\diamondsuit}$ and recover several classical comparison theorems in $p$-adic Hodge theory. We also develop analogues of our results and constructions over $\mathbb{C}_p$.