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October 8, 2025Open Access

An integral comparison of crystalline and de Rham cohomology

Key Points

The study reveals a comparison between de Rham and crystalline cohomology in a new integral context, enhancing our understanding of their relationship.
Using prismatic cohomology tools suggests new insights into torsion aspects within de Rham cohomology derived from the crystalline framework.
A stacky perspective provides a rich framework for comparing cohomology types, potentially leading to innovative methods in algebraic geometry.
This revised comparison works with coefficients from perfect complexes of prismatic F-crystals, strengthening the results of earlier studies.

Abstract

Let OK 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/OK there exists a comparison between the de Rham cohomology groups HⁱdR (X/OK) and the crystalline cohomology groups Hⁱcrys (Xₖ/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.

An integral comparison of crystalline and de Rham cohomology

Key Points

Abstract

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