It’s been a little long since I have had a chance to do more than get the pseudolectures ready – sorry! I have just trimmed the pseudolecture videos on youtube, and added brief summaries. I intend to post trimmed versions of the slides to the pseudolectures soon.

One thing I am always struck by — the very start seems to move slowly and spend a lot of time on the “basics”, and then things seem to speed up. But that is misleading — in fact we have to get comfortable with the “essentials” at the start, and then we can move ahead quickly and with confidence.

What I would like to do next (or at least, very soon) is to get back to suggesting problems for you to think about (going back a few weeks, and also going ahead a little bit, knowing that I will have weeks where it will be difficult finding time to write). I also want to spend time on zulip.

Where we left off

At the of the previous post, we were at the end of Chapter 3, which is well behind where we are in the pseudolectures. So let’s start with Chapter 4.

The first exercise of the chapter is worth doing — checking that using our definition of the structure sheaf gives the “right answer” for distinguished open sets. Roughly speaking, on \rm{Spec} \; A, the functions on the locus where f \in A doesn’t vanish should be the localization A_f where you’re allowed to invert f.

I wouldn’t skip understanding “base gluability” of the structure sheaf, which is the most complicated part of that argument. Reason: it’s not so bad, and it can sometimes be done in a horrible way, scarring people for life.

Exercise 4.3.A is also worth doing — it is enlightening for most, and strangely confusing (and particularly enlightening) for some — the fact that you can recover a ring from its spectrum in a precise way.

I would also try some of the “easy” exercises if you are feeling nervous — easy doesn’t mean unimportant. Exercises 4.3.F and 4.3.G are also important to know.

Section 4.4 has examples of schemes (and varieties). It is important to get your hands dirty and really get to know many examples of schemes/varieties — they aren’t abstract formalisms, but in fact an abstract way of understanding something concrete. You absolutely should understand projective space, and you may prefer the coordinates in the notes. The line with the doubled-origin is the ur-example of a “non-Hausdorff” space/variety/scheme.

We can get lots and lots of examples (and lots of important and historical examples) from projective geometry. Then in yesterday’s pseudolecture, I described the “Proj” construction. Graded rings and modules sound like they should be way more down to earth than fancy-schmancy things like schemes and strangely named rings, but in fact they can be more confusing! I think it is worth understanding in whatever way you like how to turn a graded ring into a “picture” (or more precisely, into some sort of geometric object).

Now that we know what schemes (and almost, varieties) are, we will define some adjectives which can be applied to them. You should think of these as either natural things you want names for, or else technical things that will turn out to be important, or else properties that essentially always hold in any reasonable situation (but that you need a name for). The first section has topological properties such as connected, irreducible, quasicompact. You’ll see quasiseparated there too — a horrible sounding name! But the thing to remember about quasiseparated is that it essentially always holds in reasonable situations, and it is also always accompanied by his little brother “quasicompact”. When we say a scheme is “quasicompact and quasiseparated” (sometimes abbreviated qcqs because it is so common a hypothesis), we just mean that it is built out of finitely many building blocks (i.e., covered by finitely many affine open sets), and their intersections are also built from finitely many building blocks (i.e. their intersections are themselves covered by finitely many affine open sets).

Then we have the geometric incarnations of “no nilpotents” (“no fuzz” = “reduced”) and “integral domain” (“integral”). Showing that integral = reduced + irreducible (5.2.F) is good practice.

In the pseudolecture, we next discussed two more stalk-local properties — normality and factoriality (section 5.4), and there are lots concrete examples to work out there (within hints!). You will notice that often an exercise that looks very geometric has next to it nearly the same exercise that looks very number-theoretic. There are a lot of exercises here to try to get your hands dirty. (I learned 5.4.H late in life, and have found it particularly useful for producing and understanding examples.)

Then we discussed the affine communication lemma (5.3), which is easy, powerful, and clever — it is worth reading and appreciating. Probably the people who will appreciate it the most are those who struggled with other ways of trying to make sense of well-definedness of definitions.

In particular, we are very close to defining varieties over a field k — they are quasicompact reduced finite-type k-schemes, that are Hausdorff. That is a strange way of saying that they aren’t built out of infinitely many pieces; they have no fuzz; they look like they are cut out by equations in k^n (or more precisely affine $n$-space); and they are (in the correct but not literal sense) Hausdorff (which we haven’t yet defined).

You should then skip the rest of chapter 5 — I have mostly rewritten the discussion of associated points/primes because it is really quite simple when done carefully, and I am unhappy with the discussion in the earlier public version you have. I’d like to get it into a shape that I can share it with you soon, because I’d get great feedback on what works and what doesn’t work. But I know that time won’t allow it.

We’ve begun to talk about morphisms of schemes (and varieties). One of the most basic insights of Grothendieck is that we shouldn’t focus on “things” (objects in a category), but instead on “maps between things” (morphisms) — properties that seem to be about objects are in fact about morphisms (they are “relative” — they are properties of a relation = map from one object to another).

So you may be able to understand fairly quickly how morphisms of schemes can be cheaply defined using the crutch of locally ringed spaces (section 6.3). Once we do that next week, we know the category of schemes; we can now talk about lots of things. The “easy” exercises around here are a great way of solidifying and verifying your understanding.

By the end of this pseudocourse, I’d like to do a good deal of chapters 7 through 10, and conclude with a large number of examples from those chapter to give you an idea of what we can now talk about, as some consolation prize instead of proving a big punchline theorem. Although the fundamental theorem of elimination theory (and elimination of quantifiers) might qualify as a fun punchline.

That’s enough for tonight! I hope to write more soon. If you would like a more explicit list of problems to think about, just let me know (in the comments below might be easiest, but other means work well too).