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

The goal of this article is to prove a comparison theorem between rigid cohomology and cohomology computed using the theory of arithmetic $\mathcal{D}$-modules. To do this, we construct a specialisation functor from Le Stum’s category of constructible isocrystals to the derived category of arithmetic $\mathcal{D}$-modules. For objects `of Frobenius type’, we show that the essential image of this functor consists of overholonomic ${\mathcal{D}}^{†}$-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 ${\mathrm{s}\mathrm{p}}_{+}$ for overconvergent isocrystals, which enables us to prove the comparison theorem for compactly supported cohomology.