Perfectoid and prismatic methods

The seminar takes place on mondays in room MA-B1-524 from 10:15 to 12:00. A discussion/exercise session takes places in the same room from 13:15 to 14:00.

I will give at least the first two sessions. Then participants are invited to present topics. Topics are listed below. There will be a priori 14 spot for talks. Talks below may be split in two or skipped if needed.

Seminar schedule

1. (7 september: Léo) Introduction to perfectoid and prismatic methods. (By Zoom!) No exercice session. Notes.

2. (14 september: Léo) Perfectoid rings, perfect prisms, Tilting equivalance. Notes.

3. (23 september: Léo) Perfectoidization, arc topology, André’s Lemma, Almost purity. Exercises this week: Tuesday 14:15, same room. Seminar: Wednesday, 10:15 same room.

4. (28 september) Perfectizations. (Aspects of p-adic Hodge theory. Section 4)

5. (5 ocotber) Perfectoid crystals and overconvergent sheaves. (Aspects of p-adic Hodge theory. Section 5)

6. (12 october) Riemman-Hilbert. (Aspects of p-adic Hodge theory. Section 6)

7. (26 october) Almost coherence and RH (Aspects of p-adic Hodge theory. Section 7)

8. (2 november) Duality and t-exactness properties (Aspects of p-adic Hodge theory. Section 9 and 10)

9. (9 november) Applications (Aspects of p-adic Hodge theory. Section 8 and 11).

10. (16 november: Léo) Bhatt Cohen-Macaullayness and applications. arXiv:2008.08070v2

11. (23 november) Applcations to MMP arXiv:2012.15801v3

12. (30 november) Survey of prismatic cohomology.

Exercises

1. Getting acquainted
2. Perfectoid rings

Bibliography

• Aspects of p-adic Hodge theory
  Bhargav Bhatt (Course Notes, Princeton/IAS)
  PDF Link

• Prisms and prismatic cohomology
  Bhargav Bhatt, Peter Scholze
  arXiv:1905.08229v4

• Absolute prismatic cohomology
  Bhargav Bhatt, Jacob Lurie
  arXiv:2201.06120

• The prismatization of p-adic formal schemes
  Bhargav Bhatt, Jacob Lurie
  arXiv:2201.06214

• Prismatic F-Gauges
  Bhargav Bhatt (Course Notes, Princeton/IAS)
  PDF Link