Seminar on Prismatic Cohomology (Spring 2025)

The goal of this seminar is to give an introduction to prismatic cohomology, a p-adic cohomology theory developed by Bhatt and Scholze. We will mainly follow [Bha18] and [BS22].

We will begin by going through the basic theory of delta rings, and use this theory to develop prisms and the prismatic site. This will allow us to define prismatic cohomology. From here, we will explore various comparison theorems, including Hodge-Tate, crystalline and étale. We hope to conclude with a more topological discussion of prismatic cohomology, particulary its relations to topological Hochschild homology (THH)

A more detailed plan of the seminar including topics for future talks and references can be found here. Please note that this document is still a work in progress.

• When : Tuesday 11:00 AM ~ 12:00 pm
• Where : Mathematics Hall, Room 528
• Organizer : Sangmin Ko and Vidhu Adhihetty
• References
  • [Bha18] Bhargav Bhatt, Eilenberg lectures on prismatic cohomology, (2018)
  • [BS22] Bhargav Bhatt and Peter Scholze, Prisms and prismatic cohomology, Ann. of Math. (2) 196 (2022), no. 3, 1135–1275. MR 4502597
  • [Ked21] Kiran Kedlaya, Notes on prismatic cohomology, (2021).

Schedule

Talk List & Abstract

Talk 1

• Title : Overview
• Abstract : We will watch a lecture given by Bhargav Bhatt at the IHES which introduces the motivation and general idea of prismatic cohomology. The lecture will importantly introduce some of the key players which will appear in later talks, such as the prismatic site, and give a rough idea of some of their features.

Talk 2

• Title : Delta rings
• Abstract : In this talk, we will define delta rings, which in spirit are rings with an analogue of p-differentiation. We will make this vague idea more precise in the talk, and also explore some of the key properties of delta rings which make them ideal for our future purposes. We will end on a discussion of p-adically complete perfect delta rings, which will be categorically equivalent to characteristic p perfect rings via Witt vectors.

Talk 3

• Title : Distinguised elements and prisms
• Abstract : TBA

Tlak 4

• Title : Perfect prisms and perfectoid rings
• Abstract : TBA