The third post is the September 25, 2010 version here. This fortnight’s reading is 4.2-5.2. At the end of it, you will be ready to know what a scheme is in general (so I’ve included 5.3 and 5.4 in case you want to read ahead, and see a few examples).

Corrections promised after the last post are included. There are some comments I still haven’t had a chance to include and digest (including from Kamal Khuri-Makdisi).

Affine schemes are the local models for schemes in general. We use them both to prove theorems (many proofs begin with “we reduce to the affine case”) and to do explicit calculations. Learners should try to get comfortable both with dealing with quite general affine schemes, and also explicit examples.

For learners.
As always, doing many exercises is essential. Which ones are right for you depend on who you are, so in this posting I’ll divide them up a little. Here are some of the more important ones.

Understanding specific examples quite explicitly will be helpful.

• You should do some of the affine space examples A^1_Q (4.2.C), A^2_C (4.2.D), A^2_Q (4.2.E), A^n_Z (4.2.M).
• 4.2.K or 4.2.L will give you practice with maps of these objects.
• 4.2.P is fun and surprising, and will give you a sense of why the dual numbers might have something to do with differentials.
• 4.6.F deals with a generic point.
• Try one of 4.6.O, 4.6.Q, and 4.7.A, which deal with reducible schemes.

There are points of theory that it is important to understand well in order to move forward. I’ve flagged them in the text.

• 4.2.F and 4.2.G tell you how primes behave under quotients and localization. Know both, and do at least one.
• Nilpotents and the nilradical are important, 4.2.N.
• The radical of an ideal is closely related, and is discussed in 4.4.D; 4.4.I and 4.5.E will be important for later, and 4.4.F is a simpler exercise if you have less time.
• 4.2.I and 4.4.G deal with maps.
• 4.7.E on how V and I are “opposites” is essential.
• 4.5.A (the distinguished base on the Zariski topology) is good practice with bases for topologies, and we’ll use it. And
5.1.A (sections of the structure sheaf over distinguished open sets) is useful, and will give you excellent practice in thinking in this new way.

Things I’d like to ask you:

• If there are examples that look like they are supposed to be easy but aren’t, please let me know. (In particular: Is 4.7.B on the equations cutting out the polynomial axes too hard given what you now know? If it is, I should remove it; or if a hint could make it easy, I’d like to add one.)
• There are two sections on visualizing schemes. These are things that are much better said in person, in conversation. How (in)comprehensible are they when reading them? More helpful to me: are there things that could be said that would help? (I realize that these sections will be more helpful to some, and less helpful to others, and I want them to be as helpful as possible to the first group, while signaling clearly to the second group that they should pass by these sections and not worry about it.)

For experts.

• Throughout the notes I rely very much on affine covers, more than some people prefer (coming up for example in the affine covering lemma, the treatment of quasicoherent sheaves, the development of cohomology). The resulting descriptions tend to be scheme-specific, and more thought is needed to extend them to ringed spaces in general. But that’s fine by me — I would prefer to do things easily, and to later see that they generalize with a little thought, than to do things more generally than is needed, and then to specialize.
• I use the phrase local ringed space rather than the standard locally ringed space. I currently feel no guilt, because “locally ringed space” can be misleading (“locally” isn’t being used in the same way we use it elsewhere — I similarly use factorial instead of locally factorial), and there is no possibility for confusion, and I tell them how the rest of the world speaks. If this really bothers you, please speak up! (And if it really doesn’t bother you, please let me know too, so you can outvote the others…)
• We’re approaching the example of projective schemes, which I must admit end up being more confusing than I had hoped.