A comparison between compactly supported rigid and $\pmb{\mathscr{D}}$-module cohomology

TL;DR

This paper establishes a comparison theorem between rigid cohomology and arithmetic $ abla$-module cohomology by constructing a specialized functor linking constructible isocrystals to $ abla$-modules, especially for objects of Frobenius type.

Contribution

It introduces a new specialization functor from constructible isocrystals to arithmetic $ abla$-modules and proves a comparison theorem for compactly supported cohomology.

Findings

Construction of a specialization functor from isocrystals to $ abla$-modules

Identification of overholonomic $ abla$-modules for Frobenius objects

Proof of the comparison theorem for compactly supported cohomology

Abstract

The goal of this article is to prove a comparison theorem between rigid cohomology and cohomology computed using the theory of arithmetic $\mathscr{D}$-modules. To do this, we construct a specialisation functor from Le Stum's category of constructible isocrystals to the derived category of arithmetic $\mathscr{D}$-modules. For objects `of Frobenius type', we show that the essential image of this functor consists of overholonomic $\mathscr{D}^\dagger$-modules, and lies inside the heart of the dual constructible t-structure. We use this to give a more global construction of Caro's specialisation functor $\mathrm{sp}_+$ for overconvergent isocrystals, which enables us to prove the comparison theorem for compactly supported cohomology.