An integral comparison of crystalline and de Rham cohomology

2025/07/23 by Abhinandan, Youcis, Alex
#14F30 #14F40 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.17631

Abstract

Let 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/O_K there exists a comparison between the de Rham cohomology groups Hⁱ_dR(X/O_K) and the crystalline cohomology groups Hⁱ_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.

Related

• Comparison of prismatic cohomology and derived de Rham cohomology
• Crystalline cohomology and de Rham cohomology
• Prismatic crystals over the de Rham period sheaf
• Exponentially twisted de Rham cohomology and rigid cohomology
• Rigid Cohomology and de Rham-Witt complexes