In the pseudolecture, I discussed more on “geometric spaces”, and at this point we have a pretty good idea of what we want if we want to make sense of something as a geometric space. We talked more about sheaves (in particular, compatible stalks, sheafification, sheaves on a base, and why using stalks we can see that sheaves of abelian groups on a given topological space form an abelian category), and we began to play around with the “local models” of the spaces we’ll discuss more (affine varieties of various sorts; and affine schemes).

Things to think about in the next couple of weeks

Make friends with some mSpec’s and Spec’s. This means go into their villages, and meet a number of them, and maybe stay over for dinner.

If you are thinking about complex analytic varieties — at this point have you fully figured out the category of complex analytic varieties?

If you are seeing sheaves on a base of a topology for the first time, can you think through why this believably has the same information as a normal kind of sheaf? If you are seeing them for the second time, I have to ask: what kind of “base” are you using? If the “usual” kind, then what does your “identity on a base” axiom look like? (Don’t look it up — I’m not asking you what’s written down by someone else!) If you are learning to think categorically (and want to — this should be done on the second pass), do you see the filtered index category lurking here?

Things to read this week

This coming week, you should be getting comfortable with everything up to the first three sections of Chapter 3, except for the last section of Chapter 1, and the last section of Chapter 2.

Problems to think about this week and next

(If the problems for different groups of people are not well-calibrated, let me know, and I’ll try to aim them better.)

For everyone: please do the same three meta-problems of what was interesting, and what was challenging, and what was confusing.

If you are new to commutative algebra:
Exercise 2.2.J might give you some practice with modules over rings. Try 2.3.C if you haven’t already. Get somewhat happy with why we can understand things about sheaves in terms of stalks, by picking a do-able problem or two in Section 2.4. Understand examples in Section 3.3 as much as you can, and practice “drawing pictures of rings”.

If you came in happy with commutative algebra:
Do 2.3.C if you haven’t already. Understand “sheaves via stalks” and “sheaves on a base” well by picking an interesting problem in each of those sections (2.4 and 2.5). Understand the examples of Section 3.3 as completely as possible.

If you are complex analytically minded:
Have you fully figured out how to think about complex analytic varieties (including morphisms between them) in the language we are using? Do you see why the fibered product of complex analytic varieties exists, for example?

If you’ve seen some commutative algebra to think about it:
Can you answer the second question I posed before the start (with rigorous proof!)? Can you describe how the maximal ideals of the polynomial ring in n variables over a field k should be identified with the Galois-orbits of n-tuples of elements of the algebraic closure \overline{k} of k?

If you have already become comfortable with the ideas we are talking about:
(This is only for those who have already seen the above, because otherwise I fear you will become a lotus-eater.) Try to mix Yoneda with “maps to a space form a sheaf”. Do this without looking up the definition of a Grothendieck topology — you should try to do this (even if you fail) without being told what to do.

Here is a precise case to think through. Suppose \mathcal{G} is the category of balls (or if you prefer, polydiscs) in \mathbb{C}^n (where n is not specified), where morphisms are holomorphic maps. Let the “functor category” ( {\text{Fun}}_{\mathcal{G}} ) of \mathcal{G} be defined by taking the objects as contravariant functors from \mathcal{G} to the category of (Sets), and morphisms are “natural transformations of functors”, so we have a (covariant!) functor Yo: \mathcal{G} \rightarrow (\text{Fun}_{\mathcal{G}} ), given by X \mapsto h_X. Two big things:

(1)(Yoneda) Yoneda’s Lemma says that this is a faithful functor, which is why we call $Yo$ the “Yoneda embedding” of $\mathcal{G}$ into its functor category (\text{Fun}_{\mathcal{G}}).

(2) (maps glue) Second, h_X is a sheaf on any Y \in \mathcal{G} (considered as a topological space).

Now \mathcal{G} sits in a bigger category, the category of complex manifolds. Show that a complex manifold X (not necessarily a ball!) gives an element of (\text{Fun}_{\mathcal{G}}), and it still satisfies “Yoneda’s Lemma for \mathcal{G}” (i.e., this element of the functor category h_X determines X up to unique isomorphism of manifolds), and also h_X is a sheaf for all Y \in \mathcal{G}.

So: figure out what it should mean for an element of (\text{Fun}_{\mathcal{G}}) to be “a sheaf” on all elements Y \in \mathcal{G}, and see what information you need to make this make sense. (Hint: you need to know when a bunch of open embeddings into some Y \in \mathcal{G} “cover” Y.) You are basically going to invent an approximation of the notion of a topology on this category (otherwise known, roughly, as a Grothendieck topology).