We are now at the end of Chapter 3, and the last week or so has been spent on “understanding the geometry of rings”, or “drawing rings”. In particular, you should now try to think of rings as topological spaces, and if you haven’t seen the topology on the spectrum (or “max spectrum”) of a ring (the Zariski topology) before, you should make friends with it, by thinking through examples. If you have seen it before, but don’t yet know how to draw nilpotents, then you can think about fuzzy pictures.

This coming Saturday, we’ll add in the sheaf of functions (or if you wish, the sheaf of algebraic functions, or regular functions), and at that point we will know about schemes, and can do some examples. Strangely, we won’t yet be able to define varieties, since we haven’t yet figured out how to describe the “Hausdorff condition”. But we’ll be close.

You will also be itching to know what morphisms of schemes are — that’s for the week after, if all goes well.

Here are some things to think about as you digest what we are talking about.

Things to think about.

Get comfortable with the examples of section 3.2. Even the case of the dual numbers is worthwhile.

Build your personal dictionary between algebra and geometry, and add to it as we go. I’ve been meaning to make a big one, and I keep starting, but it keeps getting big and then I misplace it.

Exercise 3.2.L is an example of how a geometric picture can tell you something algebraic that may not have been obvious.

Exercise 3.2.O is a hands-on example that will make certain you can see why ring maps correspond to set maps in the other direction. Exercise 3.2.P makes this rigorous and precise.

I mentioned Exercise 3.2.T in a pseudolecture, and it is just fun.

If you are new to many of these concepts: there are a number of specific algebra exercises (3.2.!, 3.2.B, 3.2.C, 3.2.G) that you can do.

If you have experience with differential geometry, try Exercises 3.1.A and 3.1.B. (This is worthwhile even if you don’t have much experience — you may get some practice with how to think about things.)

For those with more experience:
3.2.I(a) gives you one way (from Mumford?) of thinking of generic points. 3.2.I(b) can be pretty tricky.
Related: prove that the maximal ideals of k[x_1, \dots, x_n] are in some sort of “obvious” bijection with the Galois orbits of \overline{k}^n.

On the Zariski topology: there are a number of problems you should do if you haven’t seen it before. Pick problems in Section 3.4 to do. If you can, you should make them all second nature. Exercise 3.4.J has an insight that we will use repeatedly. (Exercise 3.5.E is an easy variant of it.) Exercise 3.5.B has a useful trick that comes up in ring theory a lot.

In the section on topological properties (3.6), we won’t need too much of it in depth, but it is important that you become happy with point set topology, and ideally have it digested into your unconscious. Exercise 3.6.N makes my “zen” comments about generic points somewhat more precise.

Noetherian rings turn out to be incredibly important, and it is probably a crime in several European countries that I introduced them in so little time. But there is remarkably little you need to know about them (at least to get started), and that’s what I put in the exercises in Section 3.6. It also includes statements about Noetherian modules that we won’t need (any time soon at least), but if you have seen these ideas before, you may as well make sure you’re happy with everything.

The “I(\cdot)” map from subsets of the spectra is important, and not hard. Theorem 3.7.1 and 3.7.E are the important correspondences between closed subsets and radical ideals, and between irreducible closed subsets and prime ideals, and are worth digesting. The first of these (3.7.1) is sometimes called Hilbert’s Nullstellensatz, and it is the “scheme-theoretic” version of the “variety” version stated earlier. Weirdly, the scheme-theoretic version is much easier!

Next pseudolecture

Because I may not have a chance to put up another post soon after the next pseudolecture, let me say something about what happens next. The crucial construction we will begin with is the ring of functions (if you wish, algebraic or regular functions) on the spectrum (or max-spectrum) of a ring, which we will define by instead defining it on a base — we declare that on the open subset D(f) of \rm{Spec} \; A where a function $latexf$ Doesn’t vanish, the functions are A_f — basically, we allow ourselves to divide by f. I intend to give a hands-on proof even though it is more laborious than a sneak fast proof if are happy with localizations being exact. But maybe I’ll give both. (If you want to impress your friends — this is a special case of “descent”, and the general fact underlying descent is actually just this sneaky second proof, if you look at it the right way.)

So we will then know the sheaf of functions \mathcal{O} on an “affine scheme” or “affine variety”.

This construction works without change to turn an A-module M into a sheaf \tilde{M} of \mathcal{O}-modules (the sections over D(f) are M_f, which is obviously an A_f-module).

So that means we’ll have defined what an affine scheme or variety is as a ringed space — and then we immediately know what a scheme is, or a variety (minus Hausdorffness). I would like to then do the three examples of Section 4.4 in depth, because if you understand them, you are really able to do business with schemes and varieties.

Random entertaining question for experts

(This will relate to my eventual post following up on my “category theory is central and you should never take a course on it”, retracting the last part of that statement…)

We have the notion of full functors, and faithful functors. Now fully faithful functors come up a lot. And faithful functors come up a lot. When do full functors come up? I see “full” as being a concept that mainly comes up only “after” “faithful”.

(A more dangerous question: does “essential surjectivity” of a functor come up often in cases where the functor is not already known to be fully faithful? I found it entertaining to ponder why we don’t really have a notion of “essential injectivity”…)