Here is what you can do this week, between the second and third pseudolectures. Everything here refers to The Rising Sea notes.

About the pseudolecture

The pseudolectures are to give general guides for learning things from reading and problem-solving.

Did viewing and using a single zulip stream work well (as compared to, for example, last week)? Did the half hour post-pseudolecture work well/better/fine? I felt that this was a reasonable approximation of what might happen in the future, but I am intending to tweak things more.

# Readings

Catch up on any sections before that you feel the need to look at and review.

*If you are not super-comfortable with the ins and outs of modules:* Read section 1.6, on abelian categories, stopping just before the start of 1.6.5 (on complexes). But don’t read it to learn what abelian categories; instead see why the (category of) modules over a given ring satisfy all these properties. Read the first three sections of chapter 2 as well, but starting at the bits about presheaves forming an abelian category (2.3.5), just skim.

*If you are quite comfortable with modules over a ring,* then you can read the seciton on complexes, exactness, and homology (again, thinking only of modules over a ring, not abelian categories in general), up to and including Exercise 1.6.D. Read the first three sections of chapter 2.

*If you are an expert already,* then take a look at the useful facts in homological algebra (1.6.9), and possibly continuing to the end of the section. Read the first three sections of chapter 2. Feel free to browse ahead, or to take a side trip to spectral sequences. Please quibble and argue with my point of view.

# Problems

*For everyone:*

Keep thinking about and working on the problems from last week. We can focus on them less as of next week (so you’ll have a couple of weeks to think deeply about each set of problems).

Also, please do problems 5 through 7 from last week again for this week. Please also make sure to discuss your answers in zulip so I can see them — I intend to let them guide what I say on Saturday.

5. What’s your favorite exercise (not necessarily from the notes), and why? (This is important: you are not a passive robot doing exercises. You are deliberately refining your thinking.)

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

7. What is a confusing notion you want to hear more about? (If you talk about stacks or infinity-categories, then you are showing a lack of wisdom.)

*If you are not super-comfortable with modules: * How comfortable are you with localization and tensor products? (Tensor products in particular really are confusing, so don’t be surprised, or feel stupid.) Perhaps try an exercise in each to see if you can do them. If you can, then declare victory and move on; you’ll digest these ideas better later, when you use them. Do 1.5.F (if you have to work hard, you are working too hard — t has a one-sentence answer.) Do 1.5.G (and realize that if you understand negative numbers, you understand this exercise). Do exercise 2.1.A (even though I allegedly did it in the pseudolecture). If you have some background in differential geometry, you will find Exercise 2.1.B enlightening. Exercise 2.2.B will give you an idea of when things might not quite be sheaves, but still be presheaves. Do Exercise 2.2.E, 2.2.F, 2.2.H, and even 2.2.J. Do 2.3.A, , and 2.3.C.

*If you are quite comfortable with modules over a ring: * Make sure you can do all the localization and tensor product exercises without referring to anything. Exercise 1.5.H will ensure you understand the “ification” functor. Do 1.6.A, 1.6.B, 1.6.C, and 1.6.D (for modules over a ring). Do 1.6.I, and possibly 1.6.K Do 2.1.A, and (if you know some differential or complex geometry) 2.1.B. Do 2.2.E; 2.2.F or 2.2.G; 2.2.H; 2.2.I; 2.2.J; 2.3.A. Definitely do 2.3.C. You can do 2.3.E and 2.3.F, and in anticipation of the next problem set, 2.3.I and 2.3.J.

*For experts: * If you are an expert already: do you agree with the useful facts in homological algebra (1.6.9)? Are there other things that I might not know or realize the power of, that you have found very handy to know? If you are happy with either (smooth) manifolds, or homolomorphic modules, do 2.1.B. In order to do this, you need to be clear in your own mind what your definition of cotangent space in the smooth or holomorphic setting. If you are an algebraic topological person, try the exercises on “categorical phrasing”, because you might like them. Do 2.2.F and 2.2.G, 2.2.H, 2.2.J, 2.3.A. Make sure you understand 2.3.C. Do 2.3.E and 2.3.F, and 2.3.I and 2.3.J. There is something important in these last exercises — in many places, the kernel and cokernel of maps of sheaves (of abelian groups) is defined, but it isn’t made clear *why* these are the right definitions.

# The next pseudolecture

Some of you wanted to understand “complex analytic varieties” at the same time as “algebraic varieties” and “schemes”. We’ll do that. I’ll define all of these, and smooth manifolds, in the first few minutes of the next pseudolecture, in a way that will likely infuriate you.