Once we wrapped our heads around what morphisms of schemes are like, I jumped ahead to chapter 9 to show that fibered products exist. I did this for a few reasons. I wanted to make clear that there was nothing stopping us from immediately understanding fibered products. (The reason I left it after chapter 8 in the notes is that it can take some time to digest it the first time you see it, and there was lower-hanging fruit to pick. Also, once you begin to think about the fibered product, you are led to consider many other things, so it is a substantive topic in its own right.

What I most want you to do is to listen to my exhortations about how the existence is, understood properly, “easy” (in the technical sense — it is conceptual, although you have to train your mind in order to make it natural). So watch and read and digest. Once you have digested it, you are free to read more about Yoneda’s Lemma, and “Zariski sheaves” and Grothendieck topologies — but only if you are at the stage where these are easy reads, and not when they are entrancing but opaque.

I would then recommend trying a bunch of explicit problems in section 9.2, which we haven’t discussed yet, but will let you see that you can really work with fibered products in practice. For this, you need to know something about tensor product — but you’ll find out how little there is to actually know, and how everything follows from these few facts. Section 9.3 is just about interpreting “pullbacks” and “fibers” in terms of fibered product. (Example 9.3.4, on a double-cover of the line, is super-enlightening, and I discussed it in the last pseudolecture.) And from there you can easily see why various properties are preserved by base change/pullback/fibered product (section 9.4). (Please skip 9.5, even if it would be otherwise very interesting to you — it is in the process of being rewritten.) So at this point you can plausibly be done most of chapter 9 (except for 9.5 which I asked you to skip, and 9.6, which isn’t hard, but which I’ve not yet talked about in a pseudolecture).

Here are some problems from chapter 9 that are worth trying.

If you are new to a lot of this, you can try Exercise 9.1.A, which doesn’t build on lots of other things, so you get a chance to just understand something without having to remember a huge superstructure beneath it. 9.1.B is the key connection that gets us from algebra to geometry (the “local model” of the fibered product).

On the other hand, if you are a fancy person, you can do the exercises to understand the existence of fibered products in terms of representable sheaves.

In section 9.2, I would recommend all the exercises that are the gateways through which algebra becomes geometry: 9.2.A, 9.2.B, 9.2.F

Then you can understand how to change “base fields” in this language, to for example relate things over $\mathbb{Q}$ to things over $\overline{\mathbb{Q}}$ to thinks over $\mathbb{C}$. Exercises 9.2.H to 9.2.J deal with this.

If you are a fancy person, you can try 9.2.E, which includes a ring that Jonathan Wise mentioned a few pseudolectures ago — \overline{\mathbb{Q}} \otimes_{\mathbb{Q}} \overline{\mathbb{Q}}.

And Exercise 9.2.K is not important in any way, but it is entertaining!

In Section 9.3, 9.3.A will give you some insight into fibered products — it works well for topological spaces.

In Section 9.4, if you do a few parts of problem 9.4.B, showing that various properties of morphisms are preserved by base change, then you’ll see how to do this in general.

That’s all the time I have for today — tomorrow I’ll hopefully write down a bit about classes of morphisms of schemes. I should really have done that before telling you to do Exercise 9.4.B.

I am leaning more and more toward having some “office hours” on zulip, so I can actually answer some questions you may have — I might pick a time and be there, and then I would come back to it periodically, so people can ask questions asynchronously. But only if it would be useful to enough of you (at least a few)…