Speaker: Abhinandan
Affiliation: IMJ-PRG, Sorbonne Université
Title: An integral comparison of crystalline and de Rham cohomology
Date and Time: September 3, 2025, 14:30:00 Hours
Venue: AG-77
Abstract: Let $\mathcal{O}_K$ be a mixed characteristic complete DVR with fraction field $K$ and perfect residue field of characteristic $p$. A celebrated result of Berthelot and Ogus from 1980s states that for a smooth and proper $p$-adic formal scheme $X/\mathcal{O}_K$, there exists a rational comparison isomorphism between the $p$-adic de Rham cohomology of $X/\mathcal{O}_K$ and the crystalline cohomology of the special fibre of $X$. It is then natural to ask if there exists a reasonable integral refinement of the Berthelot–Ogus comparison isomorphism and could one also incorporate coefficients integrally? In this talk, we will answer these questions by showing that such an integral refinement can be obtained by ``twisting'' these cohomology theories. More precisely, we will see that using the theory of prismatic cohomology with coefficients one can define a ``twisted'' version of de Rham and crystalline cohomology with coefficients and show that these cohomology theories agree integrally. Additionally, if time permits, in the case of constant coefficients we will also comment on the relationship between torsion in de Rham cohomology of $X/\mathcal{O}_K$ and torsion in the crystalline cohomology of the special fibre. This talk will be based on a joint work with Alex Youcis. Preprint related to the talk : https://arxiv.org/abs/2507.17631
Affiliation: IMJ-PRG, Sorbonne Université
Title: An integral comparison of crystalline and de Rham cohomology
Date and Time: September 3, 2025, 14:30:00 Hours
Venue: AG-77
Abstract: Let $\mathcal{O}_K$ be a mixed characteristic complete DVR with fraction field $K$ and perfect residue field of characteristic $p$. A celebrated result of Berthelot and Ogus from 1980s states that for a smooth and proper $p$-adic formal scheme $X/\mathcal{O}_K$, there exists a rational comparison isomorphism between the $p$-adic de Rham cohomology of $X/\mathcal{O}_K$ and the crystalline cohomology of the special fibre of $X$. It is then natural to ask if there exists a reasonable integral refinement of the Berthelot–Ogus comparison isomorphism and could one also incorporate coefficients integrally? In this talk, we will answer these questions by showing that such an integral refinement can be obtained by ``twisting'' these cohomology theories. More precisely, we will see that using the theory of prismatic cohomology with coefficients one can define a ``twisted'' version of de Rham and crystalline cohomology with coefficients and show that these cohomology theories agree integrally. Additionally, if time permits, in the case of constant coefficients we will also comment on the relationship between torsion in de Rham cohomology of $X/\mathcal{O}_K$ and torsion in the crystalline cohomology of the special fibre. This talk will be based on a joint work with Alex Youcis. Preprint related to the talk : https://arxiv.org/abs/2507.17631