Rigid cohomology of locally noetherian schemes Part 2 : Crystals

TL;DR

This paper develops a framework for overconvergent crystals on schemes, showing their cohomology aligns with de Rham cohomology and establishing descent properties that generalize previous rigid cohomology results.

Contribution

It introduces overconvergent sites and crystals, proves their cohomology matches de Rham cohomology, and establishes universal descent results in rigid cohomology.

Findings

Equivalence between constructible crystals and modules with overconvergent connections.

Cohomology of crystals is isomorphic to de Rham cohomology.

Universal descent properties for constructible crystals in the h-topology.

Abstract

We introduce the general notions of an overconvergent site and a constructible crystal on an overconvergent site. We show that if $V$ is a geometric materialization of a locally noetherian formal scheme $X$ over an analytic space $O$ defined over $\mathbb Q$, then the category of constructible crystals on $X/O$ is equivalent to the category of constructible modules endowed with an overconvergent connection on the tube $\,]X[_V$ of $X$ in $V$. We also show that the cohomology of a constructible crystal is then isomorphic to the de Rham cohomology of its realization on the tube $\,]X[_V$. This is a generalization of rigid cohomology. Finally, we prove universal cohomological descent and universal effective descent with respect to constructible crystals with respect to the $h$-topology. This encompass flat and proper descent and generalizes all previous descent results in rigid cohomology.