PrisMiPd 2023

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

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