[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.)

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.

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.

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.

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.

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.)

My supplemental notes on distinguished elements

My supplemental notes on derived completion

For a more comprehensive discussion of derived completion, we refer to the Stacks Project's excellent treatment

The first set of supplemental notes on perfect prisms

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.

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.

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.

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).

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).

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.

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.

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.

A proof is also sketched in Bhatt's Lecture 6, subject to some simplifying hypotheses.

My sketch of the proof is here.

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.