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 (note)
• 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 : Distinguished elements and prisms (note)
• Abstract : The goal of this section is to introduce the notion of prisms. To work effectively with prisms, it will be useful to define the notion of distinguished elements of a delta ring, which are essentially elements that behave as if they vanish to order 1 with respect to the p-derivation. We will also need a better notion of completions along an ideal, since the classical notion does not behave well in this setting. This issue will be solved by introducing the technical notion of derived completeness

Talk 4

• Title : Perfect prisms and perfectoid rings
• Abstract : We discuss perfect prisms, perfectoid rings and how they are essentially the same. We also give some properties of perfectoid rings.

Talk 5

• Title : The prismatic site (note)
• Abstract : We define the prismatic site and explore some of its features. We then define the prismatic complex and the Hodge-Tate complex as the cohomology complexes of certain “structure” sheaves on the site. We end with a discussion of the Hodge-Tate comparison theorem.

Talk 6

• Title : The Hodge-Tate and crystalline comparison theorems
• Abstract : We aim to prove the Hodge Tate comparison theorem. To do so, we discuss derived prismatic cohomology and how it can be used to reduce to the smooth situation. There we will discuss how crystalline cohomology relates to prismatic cohomology modulo p.

Talk 7

• Title : Derived prismatic cohomology
• Abstract : We introduce non-abelian derived functor to extend the notion of prismatic cohomology to non-smooth case and generalize Hodge-Tate comparison theorem in non-smooth case. We discuss how to compute derived prismatic cohomology by descent to quasi-regular semi-perfectoid rings, which will be useful in later talks.

Talk 8

• Title : The étale comparison theorem
• Abstract : We shall discuss mixed characteristic perfection, its relationship to the v/arc topology of a p-adic scheme, and use these themes to deduce the étale comparison theorem for prismatic cohomology.

Talk 9

• Title : q-de Rham and q-crystalline cohomology
• Abstract : We will define q-deRham complexes for “framed” smooth Z_p algebras by first defining the notion for polynomial rings, and then extending via the framing (in this context, a “framing” should be thought of as giving a set of coordinates on the smooth Z_p algebra). This construction will, unfortunately, depend on the choice of framing. However, we will show, by introducing q-crystalline cohomology, that the above construction is independent of the framing when one passes to the derived setting.

Talk 10

• Title : Kodaira Vanishing - Part I
• Abstract : In the first of the two talks on mixed characteristic Kodaira vanishing, we will introduce the main theorem as an analogue of the classical Kodaira vanishing theorem. We will also explain how it relates to Cohen-Macaulayness of absolute integral closures of rings, and lay out the proof the new Kodaira vanishing theorem. A serious input will be Bhatt and Lurie’s p-adic Riemann-Hilbert correspondence. In this talk, we will work modulo this correspondence, and leave discussions of p-adic Riemann-Hilbert for the second talk.

Talk 11

• Title : Kodaira Vanishing - Part II
• Abstract: In the previous talk we discussed Bhatt’s elegant theorem and saw glimpses of just how deep the theorem is via the complicated theories that were used to prove it. Our goal now is to describe in more detail these theories i.e. log prismatic cohomology and the Riemann-Hilbert correspondence in mixed characteristic, with the goal of further filing in the skeleton of Bhatt’s proof.