Happy new year everyone! I expect to get back to posting every two weeks, and I am going to start to catch up with the backlog of comments to respond to.

The ninth post is the January 8 version here, ending with 14.1. Section 13.4 on valuative criteria is included, but I haven’t edited it yet in response to the useful comments here, and hope to do so by the next posting.

For learners.

Read 12.3, 13.1, 13.2, and 13.3. Skip 13.4 (because I haven’t edited it). Likely skip starred sections 12.4 and 13.5 (depending on your taste), and feel free to read ahead to 14.1.

12.3 is on a couple of very useful codimension one theorems (Krull and Hartogs). It is most important to get comfortable with their statements and applications. The proofs are less essential.
Starred section 12.4 proves Krull’s theorem, and introduces Artinian rings.

13.1 is intended to define the Zariski tangent space, and to show you that it is easy to work with.
13.2 defines the algebro-geometric version of smoothness.
13.3 is on discrete valuation rings (“smoothness in codimension 1”), which prove to be of critical importance.
You should skip 13.4 on valuative criteria (because I haven’t edited it), and likely also the starred section 13.5, on completions (power series).
14.1 is an introduction to vector bundles and locally free sheaves. There isn’t much content, but there is an important and subtle change in perspective from what you may be used to. That’s why I wanted you to take a look at it.

The ten exercises you should most do:
12.3.C, 12.3.E, 13.1.B, 13.1.E, 13.1.G, 13.2.B, 13.2.E, 13.2.J, 13.3.A, 13.3.G.

Exercises involving explicit examples that you should sample: 12.3.B, 13.1.C, 13.1.D, 13.1.H, 13.2.C, 13.2.F, 13.2.H, 13.2.K, 13.3.B.

If someone tries 12.3.G (successfully or unsuccessfully), please tell me. I want this exercise to be do-able, and I’ve added a hint that I hope might help. (An earlier version was too hard.)

You may find it worthwhile to read section 12.4 on the proof of Krull’s Principal Ideal Theorem, because it will introduce you to Artinian rings. If you try and find something confusing, please let me know.

For experts.

The (optional) proof of Krull gives an opportunity to introduce Artinian rings and modules, which frankly don’t enter the story later in the notes (despite of course being important “later on” in much of the field). So this is a good place to let the most interested of readers know the main facts, which they should be able to prove themselves. Question: What the facts about Artinian rings people should most know? The ones I can think of (that aren’t currently there) are the following. Artinian rings are Noetherian. Artinian rings over a field are finite type. A finite type k-ring is Artinian iff it is dimension 0. Anything else? (And is there a proof of Krull that you like better?)

My sense is that when you learn algebraic geometry, and you know that you like the idea of smoothness, you want A^n_k to be smooth. But it is a hard fact that this is true (for n at least 3). My current position is that one really doesn’t care about regularity in general (at this stage in learning, for most people). Regularity in dimension 1 is of course important.

What is the right definition of node (of a curve), cusp, or tacnode? Currently I give the definition over an algebraically closed field: it should be a point of a finite type scheme, of dimension 1, whose completion looks as you expect. This seems a bit clunky, but I can’t think of anything better. I’ve chosen to duck the question over a general field, but perhaps I shouldn’t. (I would define it by base changing to the algebraic closure. But I’d prefer to avoid issues such as the fact that the point can split into several points, and then one is a node if the others are. Readers interested in such things should probably think through this themselves. Readers less interested will be distracted — and many complex-minded readers will be very interested in the notion of node, cusp, etc., and I don’t want to make it seem harder than it is for them.)