The second post is the September 10, 2010 version here. This fortnight’s reading is until the end of 4.1, although I’ve included 4.2 in case you want to read ahead.

Corrections promised after the last post are included.

General comments. Sheaves are fundamental, and are harder than we (as teachers) remember. It is tempting to get through them as quickly as possible to get to the real stuff. This is dangerous — later in the material, it becomes clear that many students don’t think fluently about them. There is another advantage of taking time to learn about sheaves: students who leave a class before getting deeply into schemes take something valuable away with them. (Similarly, if they stick with the class a little longer, they begin to absorb the valuable lesson that a geometric space is well thought of as a topological space with a sheaf of functions, and that we really understand spaces via their functions.)

For learners. Read 3.1 to 4.1. Take your time to absorb the idea of sheaves. As with category theory, it takes a little time to learn how to think in this way. The real test of your understanding is your ability to solve problems.

If you would like a “problem set”, here is hard one: 3.2.F (morphisms glue), 3.2.H (pushforward sheaf), 3.3.B (sheaf Hom), 3.3.E (cokernel presheaf), 3.4.E (isomorphisms are determined by stalks), 3.4.J (construction of sheafification by compatible germs), 3.4.O (on surjection of sheaves), 3.5.D (stalkification is exact), 3.5.E (taking global sections is left-exact), 3.5.I (tensor products), 3.6.B (adjunction of inverse image and direct image), 3.6.G (support), 3.7.B (on sheaves on a base). But you should do some others as well (and perhaps instead). If you can do almost all of these, plus some more you find interesting, you are in excellent shape.

And if you have even a little experience with manifolds, try some of the problems in 4.1.

Something else you should do: decide which exercises you like best, and which you like least, and why. Don’t just be told what to do — develop your own taste.

For experts.
Here are things that are tricky the first time you learn them: how even to think about a sheaf, a notion which contains so much information; the fact that you can check various things on stalk; sheafification, and how to work with it; the inverse image sheaf; sheaves on a base.

I first try to convince them that they already know the idea of a sheaf, in that they know how continuous (or differentiable, etc.) functions behave.

Some smaller less important comments and questions.

(a) One axiom sometimes used in the definition of a sheaf is that F(\emptyset) is the final object in the category. This follows from the other axioms, once the empty product is defined appropriately.

(b) What is reasonable notation for a skyscraper sheaf (say with set S, at point p in the space X)? I’m using S_p, but this conflicts with stalk notation.

(c) I don’t like the phrase “constant sheaf”, because sections are just locally constant. I thus prefer “locally constant sheaf”. I realize that this conflicts with the sheaf of sections of a finite etale cover, but this doesn’t bother me; both are locally constant sheaves, just in different topologies. Is it so terrible to use the phrase “locally constant sheaf” instead of the conventional “constant” sheaf?

(d) The cokernel sheaf is often just defined. But one needs to know that it deserves this name, i.e. satisfies the universal property of the cokernel. This is a bit annoying, but it is also a bit enlightening. Similarly for the image sheaf.

(e) I have no great insightful explanation of why the inverse image (defined as a colimit) should be adjoint to pushforward.

(f) The Grothendieck quote at the start of Ch. 5 is Colin McLarty’s translation.

(Note to self: 14 comments read. No loose ends.)

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