Since the last update we have added the following material:

- A chapter on crystalline cohomology. This is based on a course I gave here at Columbia University. The idea, following a preprint by Bhargav and myself, is to develop crystalline cohomology avoiding stratifications and linearizations. This was discussed here, here, and here. I’m rather happy with the discussion, at the end of the chapter, of the Frobenius action on cohomology. On the other hand, some more work needs to be done to earlier parts of the chapter.
- An example showing the category of p-adically complete abelian groups has kernels and cokernels but isn’t abelian, see Section Tag 07JQ.
- Strong lifting property smooth ring maps, see Lemma Tag 07K4.
- Compact = perfect in D(R), see Proposition Tag 07LT.
- Lifting perfect complexes through thickenings, see Lemma Tag 07LU.
- A section on lifting algebra constructions from A/I to A, culminating in
- Elkik’s result (as improved by others) that a smooth algebra over A/I can be lifted to a smooth algebra over A, see Proposition Tag 07M8.
- Given B smooth over A and a section σ : B/IB —> A/I then there exists an etale ring map A —>A’ with A/I = A’/IA’ and a lift of σ to a section B ⊗ A’ —> A’, see Lemma Tag 07M7.

- We added some more advanced material on Noetherian rings; in particular we added the following sections of the chapter More on Algebra:
- Some results on power series rings is a short technical section on properties of power series rings over fields and Cohen rings,
- Permanence of properties under completion contains a discussion of properties of local rings which are preserved under completion,
- Permanence of properties under henselization contains a discussion of properties of local rings which are preserved under henselization,
- Filed extensions, revisited talks about p-bases,
- The singular locus talks about the singular locus of Spec of a Noetherian ring,
- Regularity and derivations shows that the existence of derivations

sometimes helps to prove rings are regular, - Formal smoothness and regularity shows that A —> B is formally smooth (in m-adic topology) if and only if A —> B is regular (due to Andr\’e IIRC),
- G-rings contains generalities about G-rings,
- Excellent rings introduces them and doesn’t do much else besides.

- You’re going to laugh, but we now finally have a proof of Nakayama’s lemma.
- We started a chapter on Artin’s Axioms but it is currently almost empty.
- We made some changes to the results produced by a tag lookup. This change is a big improvement, but I’m hoping for further improvements later this summer. Stay tuned!
- We added some material on pushouts; for the moment we only look at pushouts where one of the morphisms is affine and the other is a thickening, see Section Tag 07RS for the case of schemes and see Section Tag 07SW for the case of algebraic spaces.
- Some quotients of schemes by etale equivalence relations are schemes, see Obtaining a scheme.
- We added a chapter on limits of algebraic spaces. It contains absolute Noetherian approximation of quasi-compact and quasi-separated algebraic spaces due to David Rydh and independently Conrad-Lieblich-Olsson, see Proposition Tag 07SU.

Enjoy!

The last result mentioned will allow us to replicate many results for quasi-compact and quasi-separated algebraic spaces that we’ve already proven for schemes. Most of the results I am thinking of are contained in David Rydh’s papers, where they are proven actually for algebraic stacks. I think there is some merit in the choice we’ve made to work through the material in algebraic spaces first, namely, it becomes very clear as we work through this material how very close (qc + qs) algebraic spaces really are to (qc + qs) schemes.

Hi. Regarding the formal smoothness stuff (I’ve been trying to work through some of this myself), I’m a little confused about the role of “topologies” in it. For instance, EGA proves the following result:

Theorem: Let $latex k$ be a field, $latex A$ be a noetherian local ring which is a $latex k$-algebra. Then $latex A$ is formally smooth over $latex k$ if and only if it is geometrically regular.

Is there any result like this when we put on $latex A$ the discrete topology? I guess I’d like to keep everything discrete. In particular, I’m curious if the following is true:

Let $latex A \to B$ be a morphism of noetherian rings. Then $latex B$ is a formally smooth $latex A$-algebra if and only if $latex B$ is flat over $latex A$ and $latex B \otimes_A k(\mathfrak{p})$ is geometrically regular as a $latex k(\mathfrak{p})$-algebra for each prime of $\latex A$.

Lemma Tag 07EC says that if A —> B is formally smooth for the discrete topology, then it is formally smooth in any adic topology you put on B. (The converse is not true.) Hence the final statement of your comment is true.

Sorry, I was too quick in answering this, see comment below.

Actually, maybe in view of Popescu’s theorem (http://www.ams.org/journals/jams/1999-12-02/S0894-0347-99-00294-5/S0894-0347-99-00294-5.pdf) it’s not a realistic expectation? Is there a simple example of a filtered colimit of smooth morphisms which is not formally smooth?

Please do not refer to Spivakovsky’s paper for this theorem, since I think nobody checked his arguments carefully and in detail. The introduction to the chapter on Popescu’s theorem tries to give some overview of the literature.

As to your math question: It is easy to show that if A —> B is a homomorphism of *Noetherian* rings and B is a filtered colimit of smooth A-algebras, then A —> B is regular. See Lemma Tag 07EP.

Sorry, I now realize that you are thinking about the implication in the other direction, i.e., if A —> B is regular, then is A —> B f.s. in discrete topology. I don’t think so. Namely, A –> B is formally smooth implies that Omega_{B/A} is projective. But I think that for example Omega_{k[[x]]/k} (not continuous!) is not a projective k[[x]]-module.

Ah, very interesting. I’ll think about this.

I guess you do need some kind of finiteness hypotheses then. I think I understand the proof that formal smoothness implies a bunch of things, but going in the other direction seems harder or, in this case, impossible.

I’m curious: Is the “More on Algebra” chapter suppose to make the “Algebra” chapter of a reasonable length or is it a separate chapter because some algebra are easier using the derived category?

It started out as “a separate chapter because some algebra are easier using the derived category” but then it became a place to put more advanced commutative algebra material (partly because the commutative algebra chapter is too long already).

So every now and then I think about the question: “What is a good way to split the CA chapter in 2?” Recently I have been thinking that we could simply split it using “length of proof” (measured by the longest chain of references reaching the result). In that way we could have “Introductory algebra”, “Commutative Algebra”, and “Advanced Algebra” (or some such titles). The problem with that approach is that it isn’t very “thematic” and you’ll have random lemmas all over the place in the first chapter…

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