Prismatic Cohomology

Spring 2020

Course Information

Since the class is happening remotely, it will be closer to a group reading course, with the scheduled class time serving as a discussion section.
[Note: in the end, this didn't happen --- the lectures just worked through the various sets of lecture notes, with occasional sojourns into the original source papers.]
The core reading will be Bhargav Bhatt's lecture notes, which we will follow closely. I will also upload supplemental notes from time to time.
We will meet over Zoom from 12:30 to 1:50 on Tuesdays and Thursdays; the invitations will be sent via the class mailing list.
(Please email me if you are not on the list and would like to be --- although I can't promise to open the course to non-U-of-C participants.)

Overview

Prismatic cohomology is a recently developed cohomology theory for schemes over p-adic rings. It is related to both (p-adic) etale and de Rham/crystalline cohomology, and so also to p-adic Hodge theory.
It relates to aspects of p-adic Hodge theory that previously had not seemed to be of a cohomological nature --- for example, Breuil--Kisin modules, which previously had not appeared naturally in the geometric
(as opposed to Galois theoretic) parts of p-adic Hodge theory, are a natural output of prismatic cohomology.

References

Prisms and Prismatic Cohomology, by Bhatt and Scholze. (This is the core reference)
Integral p-adic Hodge Theory and Topological Hochschild Homology and Integral p-adic Hodge Theory, by Bhatt, Morrow, and Scholze.
(These are key precursors to the prismatic theory, although, due mainly to my own limited understanding, I don't plan on discussing them much during the course.)
Prismatic cohomology lecture notes by Bhatt. These are the main reference for the course.
These collections of notes from Winter and Spring of 2019, written by Beilinson and Drinfeld, give a different perspective on the prismatic theory. I would
very much like to understand this, but we may not get to it in the course.

Additional remarks

Prismatic cohomology has more of a categorical/homotopical algebra flavour than I am used to dealing with (e.g. cotangent complexes play an important role,
and simplicial commutative rings come in to play). As this material arises, I may add discussions of some of it as part of the supplemental material. This material
may be a bit haphazard, since it is based on my own background knowledge and tastes, but if people would like to see particular background material explained,
I can try to add material explaining it.

Schedule

Here is the schedule of lectures (in progress, and aspirational --- there will likely be modifications as we see how the quarter progresses).

Lecture zero --- Introduction

Bhatt's Lecture 1
My own introduction/motivation

Please read these in advance of our first meeting. If you don't have one already, these will give you a sense of why prismatic cohomology is important
and interesting. As I explain in my introductory note, part of my own interest in prismatic cohomology comes from thinking about the paper
Mod p Hecke operators and congruences between modular forms of Ribet.

Lecture one --- Delta rings

Bhatt's Lecture 2
My supplemental notes on delta rings

The forgetful functor from delta rings to rings is interesting --- it has both a left and right adjoint, and so preserves both limits and colimits.
There is an interesting categorical formalism related to adjoint functors, namely the theory of (co)monads, which I found useful for thinking
about this situation. To help with this, I wrote up some notes on adjoints and (co)monads.

I found these online sources helpful when writing my notes:
An online copy of Category Theory by Steve Awodey ( Chapter 10 espeically)
Notes (by David Mehrle) of lectures by Peter Johnstone on Category Theory (especially Chapter 5)
The Stacks Project also has a useful succinct treatment of the Adjoint Functor Theorem

Here is a useful repository of articles on Witt vectors. (The functor of p-typical Witt vectors provides the right adjoint to the forgetful functor from delta rings to rings.)

Lecture two --- Distinguished elements

Bhatt's Lecture 3
My supplemental notes on distinguished elements

Lecture three --- Prisms

Bhatt's Lecture 3
My supplemental notes on derived completion
For a more comprehensive discussion of derived completion, we refer to the Stacks Project's excellent treatment

Lecture four --- Prisms (continued)

My supplemental notes on prisms
The first set of supplemental notes on perfect prisms

Lecture five --- Perfect prisms

Bhatt's Lecture 4
The continuation of my supplemental notes on perfect prisms, explaining the equivalence between perfect prisms and perfectoid rings.
The preceding notes inadvertently omitted a proof that perfectoid rings (defined as A/I for a perfect prism (A,I)) are automatically classically p-complete. These notes give the proof.
Also, there is a simple delta ring identity, explained here, which is easy to remember and understand, and which simplifies some of the other elementary arguments with delta rings that we've been making.

Lecture six --- Finishing perfect prisms, and examples

Here are some examples.

Lecture seven --- Delta rings and divided powers

The relationship between delta rings and divided powers is fundamental. It plays a key role both in comparing prismatic cohomology for crystalline prisms to crystalline cohomology, and in proving the Hodge--Tate comparison in general (via reduction to the crystalline case). The basis of this relationship is an elementary lemma relating delta rings to divided power envelopes, which (along with some other, related, results) is proved here.
This lecture and the next expand on Lemma 2.1 in Bhatt's Lecture 6, which itself encapsulates the discussion of Section 2.5 in the Bhatt--Scholze paper.

Lecture eight --- More on delta rings and divided powers

We continue discussing the relationship between delta-rings and divided power envelopes. Here are the notes.

Lecture nine --- Introduction to simplicial methods

Arguments with simplicial commutative rings play an important role in the prismatic story (and in other recent developments in number theory, such as derived deformation theory). The next couple of lectures are aimed at providing a rapid introduction to these ideas.
Here is the first set of notes, which begins with simplicial sets and builds up to some ideas about simplicial abelian groups.
Akhil Mathew's notes on the Dold--Kan correspondence are a very useful introduction to this circle of ideas.
Of course, the Stacks Project also cover this material, in its chapter on simplicial methods.
Jacob Lurie's Higher Topos Theory is the basic reference for infinity categorical ideas, and also explains a wealth of related homotopical background. Kerodon is the corresponding go-to online source.

Lecture ten --- Simplicial methods (continued)

This lecture will continue the discussion of simplicial methods. (We only got about half-way through the notes last time.)

Lecture eleven --- Simplicial commutative rings

I found Akhil Mathew's notes on simplicial commutative rings very helpful.
Here is the beginning of my own notes on simplicial rings; they primarily explain how to think of derived tensor products, and derived completions, as simplicial rings (or, better, as animated rings), rather than just complexes.
These notes explain how to interpret a delta-ring structure as a derived Frobenius lift (following Bhatt--Scholze; see Remark 2.5 of their paper and Remark 3.4 of Bhatt's Lecture 2).

Lecture twelve --- Preview of what's next

This lecture gave an overview of some of the results to come in Bhatt's Bhatt's Lecture 5 and Bhatt's Lecture 6. There are no notes. (Sorry!)

Lecture thirteen --- The Cartier isomorphism

The theory of the Cartier isomorphism is briefly summarized in Construction 1.9 of Bhatt's Lecture 6.
My notes give a more detailed background and explanation.

The famous paper of Deligne and Illusie on degeneration of the Hodge-to-de Rham spectral sequence also provides background on the Cartier isomorphism, while Serre's classic paper on the topology of algebraic varieties in characteristic p develops the basic theory of the Cartier isomorphism in the case of curves (see Sections 10 and 11; note that the proof of the connection between the Cartier operator on holomorphic differentials and the Hasse--Witt matrix that I give in my notes is essentially the same as the one Serre gives here; also, Section 12 discusses the case of elliptic curves which is also discussed in my notes).
According to Serre, the proof that a 1-form fixed by the Cartier operator is necessarily logarithmic (stated without proof in my notes; see also p. 41 of Serre's paper) goes back to Jacobson; see Theorem 15 of his paper Abstract derivation and Lie algebras (reference [17] of Serre's paper).

Lecture fourteen --- An overview of sites and Cech formalism

Here are the notes.

Lectures fifteen and sixteen --- Introduction to crystalline cohomology

There are many references for crystalline cohomology. Bhatt gives a rapid summary in his Lecture 6. Marc-Hubert Nicole has a very nice, somewhat more detailed, summary in these notes.
My notes provide another summary, somewhere between the two previous references in their level of detail.
The paper Crystalline cohomology and de Rham cohomology of Bhatt and de Jong gives a quite direct proof of the relationship between crystalline cohomology and de Rham cohomology. It uses some topos-theoretic language, but in a fairly gentle way, and so can also help as an introduction to the topos-theoretic view-point.

Lecture seventeen --- Complete flatness and related topics

The notion of complete flatness is discussed in Bhatt--Scholze, in the introduction and again, in a more simpicial context, in Section 2.6. It is used in Section 3 in Lemma 3.7 (to analyze properties of modules over perfect prisms) and subsequenty in the key Proposition 3.13, which concerns prismatic envelopes.
It is briefly discussed in Section 2 of Bhatt's Lecture 2.
These notes introduce and discuss the basic properties of complete flatness, including the simplicial aspects used in Bhatt--Scholze. They also include the proof of Lemma 4.7 of Bhatt, Morrow, and Scholze's second paper (which is cited in the proof of Lemma 3.7 of Bhatt--Scholze), as well as a brief discussion of the notion of ``complete relative regularity'' of a sequence --- a notion which appears in the statement and proof of Bhatt--Scholze's Proposition 3.17.
These preliminary notes discuss some aspects of simplicial ring theory (including the construction of Koszul complexes as simplicial rings) that are used in the previous notes.

Lecture eighteen --- Prismatic envelopes

With our new-found comfort regarding simplicial methods, we'll tackle the proof of Proposition 3.13 in Bhatt--Scholze.
This is Corollary 2.3 of Bhatt's Lecture 6, but the proof is only sketched there.
We first need yet another result related to the comparison between divided power envelopes and certain delta-ring constructions that we made above. This is Corollary 2.43 of Bhatt--Scholze, and a proof is also sketched here.
The proof of Proposition 3.13 itself is the subject of these notes.

Lecture nineteen --- Crystalline comparison

The crystalline comparison theorem is proved as Theorem 5.2 of Bhatt--Scholze.
A proof is also sketched in Bhatt's Lecture 6, subject to some simplifying hypotheses.
My sketch of the proof is here.

Lecture twenty --- Hodge--Tate comparison

This is proved in sections 5 and 6 of Bhatt--Scholze.
A proof is also sketched in Bhatt's Lecture 6, again subject to some simplifying hypotheses.
My proof sketch is here.
The essence of the proof is very simple --- one reinterprets the inverse Cartier isomorphism in prismatic terms, via the crystalline comparison theorem, and then uses base-change arguments to extend from the case of crystalline prisms to the case of general bounded prisms. The results on prismatic envelopes ensure that the cosimplicial rings involved in computing prismatic cohomology (by passing to total complexes) are compatible with base-change. The main technical issue in the proof is that totalization and (derived) base-change may not commute, since derived tensor products are left-derived, while the total complex associated to a cosimplicial ring or module is bounded below, but not necessarily above. So one first proves base-change results under certain finite Tor-amplitude assumptions, which turn out to be sufficient for establishing the Hodge--Tate comparison. General base-change then follows (see Corollary 4.11 of Bhatt--Scholze).
In characteristic 2 there is also a technical issue that arises in defining the Hodge--Tate comparison map, related to commutativity vs. strict commutativity for the cup product on prismatic cohomology. This is resolved in Bhatt--Scholze by the same sort of base-change arguments as go into the proof of the Hodge--Tate comparison (see there Proposition 6.2). A more direct proof, due to de Jong, appears as Lemma 5.4 in Bhatt's Lecture 5.