An integral comparison of crystalline and de Rham cohomology

TL;DR

This paper extends the classical comparison between de Rham and crystalline cohomology for smooth proper schemes over mixed characteristic DVRs by using prismatic cohomology and stacky methods, revealing new insights into torsion phenomena.

Contribution

It introduces a prismatic cohomology-based comparison with coefficients in perfect complexes, offering an integral perspective and new tools for understanding torsion in cohomology theories.

Findings

Established a prismatic comparison with coefficients in perfect complexes.

Provided new methods to analyze torsion in de Rham and crystalline cohomology.

Enhanced understanding of the integral relationship between these cohomology theories.

Abstract

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 $X/\mathcal{O}_K$ there exists a comparison between the de Rham cohomology groups $\mathrm{H}^i_\mathrm{dR}(X/\mathcal{O}_K)$ and the crystalline cohomology groups $\mathrm{H}^i_\mathrm{crys}(X_k/W(k))$ 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.