An integral comparison of crystalline and de Rham cohomology

Let ${\mathcal{O}}_K$ be a mixed characteristic complete DVR with perfect residue field $k$ and fraction field $K$. It is a celebrated result of Berthelot and Ogus that for a smooth proper formal scheme $X\mathrm{/}{\mathcal{O}}_K$ there exists a comparison between the de Rham cohomology groups ${\mathrm{H}}_{\mathrm{d}\mathrm{R}}^i(X\mathrm{/}{\mathcal{O}}_K)$ and the crystalline cohomology groups ${\mathrm{H}}_{\mathrm{c}\mathrm{r}\mathrm{y}\mathrm{s}}^i(X_k\mathrm{/}W(k))$ of the special fibre, after tensoring with $K$. In this article, we use the stacky perspective on prismatic cohomology, due to Drinfeld and Bhatt–Lurie, to give a version of this comparison result with coefficients in a perfect complex of prismatic $F$-crystals on $X$. Our method is of an integral nature and suggests new tools to understand the relationship between torsion in de Rham and crystalline cohomology.