PrisMiPd 2023

Program

The goal of this workshop is to study Prismatic Cohomology. First we will focus on the foundational paper by Bhatt and Scholze "Prisms and prismatic cohomology" where prismatic cohomology is defined as sheaf cohomology for the prismatic site. In loc. cit. several properties of this cohomology are studied, in particular the comparison with crystalline, étale and $\mathbb{A}_{inf}$-cohomology. Secondly we will focus on the stacky approach introduced by Drinfel'd in "Prismatization" and Bhatt-Lurie in "Absolute prismatic cohomology"

Locandina

First Meeting (January 18, 2023 Milan)

Goal: review the article [BSppc] and undertsand the main comparison theorems: i.e. how prismatic cohomology is related to de Rham, crystalline and étale cohomology.

  • 1. Delta rings and prisms - by A. Bertapelle
    [BSppc: \S 2,3] State Theorem 1.8. Explain \S 2: in particular 2.11, 2.20, 2.32, 2.35. Give a shor account on prisms: in particular 3.2 and 3.9.
    10:00-11:00

  • 2. The prismatic site and comparisons (crystalline/de Rham) - by L. Fiore
    [BSppc: \S 5,6]
    11:15-12:15

  • 3. Perfectoidization - by M. Seveso
    [BSppc: \S 7,8]
    14:00-15:00

  • 4. The étale comparison theorem - by R. Venerucci
    [BSppc: \S 9]
    15:15-16:15

  • 5. Almost purity - by F. Andreatta
    [BSppc: \S 10]
    16:30-17:30

Second Meeting (February 8, 2023 Padova, Room 1BC50)

Goal: define the Cartier-Witt stack and its relation with prismatic cohomology

  • 1. On de Rham cohomology theory via stacks - by J. Scholbach
    [Bhatt video; Dp \S3] Describe the ring stacks related to de Rham. The concept of de Rham space should be discussed. Also the notion of the stack "Cone(d)" obtained from a map of group schemes should be explained.
    10:00-11:00

  • 2 Stacky approach to crystals - by A. Vezzani
    [Dsac] Explain theorem 2.4.2 stating the equivalence between the category of crystals on X and the category of qcoherent modules over a certain stack. Here X is a frobenius-smooth scheme over FF_p
    11:15-12:15

  • 3. Prismatization 1 - by N. Mazzari
    [Dp; BLapc \S 3.1-3.3; Dnrg; Dsac \S3] Construct de Prismatization of ZZ_p. This is denoted \Sigma by Drinfeld and WCart by Bhatt-Lurie. See also this pdf of [PS]
    14:00-15:00

  • 4. Prismatization 2 - by F. Binda
    [Dp; BLapc \S 3.4-3.5] Prismatization in general or the properties of the Cartier-Witt stack. This pdf of [PS]
    15:15-16:15

  • 5. Some results - by M. D'Addezio
    Sen theory [BLapc \S3], Absolute prismatic cohomology [BLapc: \S 4], Diffracted de Rham cohomology
    16:30-17:30