The trimmed version of pseudolecture 3 should be posted before long. I hope you have been able to find them!

We are now moving forwards on three fronts at once:  sheaves, ringed spaces, and varieties/schemes.

This week, we made better friends with presheaves of abelian groups on a topological space (and showed that they form an abelian category, by working “open set by open set”, and began to make better friends with sheaves of an abelian group (by thinking of them in terms of stalks).  This week I’d like you to get comfortable with thinking of sheaves in terms of compatible germs, and in particular begin to understand sheafification.    (And yes, the compatible germs is “in essence” the same as the espace etale construction, so if it makes you happier, think of it in this way.)

We also began to understand geometric spaces as (locally) ringed spaces.  More specifically, we defined a number of them as ringed spaces (including smooth manifolds, complex manifolds, and complex analytic varieties).   But although we know the objects of these categories, we haven’t figured out what morphisms should be.

Finally, we began to get at varieties and schemes by defining the underlying set of our “local building blocks”, mSpec and Spec respectively.

So the readings (to do this week) and problems (to ponder this week and next) are intended to make progress on each front.

For everyone, please do the three “standard” problems, and report on the answers in zulip (perhaps as a working group if you prefer, and otherwise individually is great).

A. What’s your favorite exercise (not necessarily from the notes), and why?

B. What was a big insight here (either new to you, or perhaps not), and why?

C. What is a confusing notion you want to hear more about?

If you are still getting comfortable with modules

Get solid on Chapter 2 up until 2.3. Try Exercise 2.3.J. Read sections 2.4 and 2.5. I’m not sure how hard you will find these sections, as it doesn’t involve much algebra, but does involve geometric intuition. Try 2.4.A, 2.4.B, 2.4.C. 2.4.E is worth doing, as you’ll see “how sheaves are better”. 2.4.F is good to practice earlier idea. You might even like to try the exercise describing sheafification (2.4.H through 2.4.J). Try 2.4.O too. (And of course, try others if you can.) In 2.5, think through 2.5.A (no need to write it up, but just convince yourself.) See if you can do one of the important exercises here.

Are you yet convinced of the notion of a sheaf as an important object? (Perhaps not yet — but at least think it over.)

Regarding geometric spaces as ringed spaces — ponder this, and see if you can come to terms with it, or at least try to say explicitly what is confusing about it. If you’ve seen some differential geometry, do 3.1.A and 3.1.B. Read 3.1.

And for the underling set for our “building blocks” for varieties and schemes: read 3.2 up until 3.2.I closely, and think through the examples as completely as possible, and digest them as best you can. It is written for schemes, but instead pretend that Spec is replaced by mSpec, and that the ring you are considering is finitely-generated over the complex numbers. Then depending on your background, generalized th finitely-generated algebras over an algebraically closed field; or over a field; or do Spec’s and rings in general.

If you are quite comfortable with modules over a ring

Read chapter 2 up to 2.5. Do 2.3.C if you haven’t already. Do 2.3.F and 2.3.G. Do 2.4.A, 2.4.B, 2.4.C, 2.4.D. Work out sheafification completely, in particular doing 2.4.J. Do 2.4.K and 2.4.L. 2.4.M and 2.4.N, and see how they are difference. And 2.4.O is enlightening! In 2.5, try a couple of exercises so you see how the idea works — 2.5.C is good. 2.5.D might be brain-bending, or you might see how to do it quickly and easily. 2.5.E may be enlightening for you.

In chapter 3, read 3.1 an, and 3.2 up until 3.2.I. If you have seen some differential geometry, do 3.1.A and 3.1.B. In 3.2, understand the examples as completely as you can, and also the variants where Spec is replaced by mSpec.

If you are an expert, or are very comfortable already with the ideas here

I’m looking for places where you disagree with me, or are surprised by something. I find it interesting which results about “things determined by stalks” are hard and which are not. Is there a better way to the equivalence of categories of sheaves on a base, and sheaves? In chapter 3, I find the description of the cotangent space is confusing to people not because the definition is confusing, but because people sometimes are thinking too fuzzily about tangent spaces in the old-fashioned way.