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.