Toy sheaf cohomology computation

I asked this question a while back on MO :

One thing that really helped in learning the Serre SS was doing particular computations (like $H^*(CP^{\infty})$)

I am curious, as a sort of followup if anyone can suggest:

1. A reference where small computations are carried out? or
2. A specific computation to do with a small enough sheaf an some simple topological space that would be able to give one a feel for sheaf cohomology. So this space that we are working over need not be a scheme, in fact it would probably be best if it were not a scheme since I don't understand them quite yet. And are there tricks of the trade to computing these things? or do people just hammer away ate injective resolutions?

In short, please suggest a space and a sheaf on it that I should work on computing the sheaf cohomology of.

PS: I of course welcome any other suggestions for understanding how to compute sheaf cohomology.

Solution 1:

Any de Rham cohomology (or Dolbeault cohomology) computation is a computation in sheaf cohomology. Actually --- any computation in singular cohomology is a computation in sheaf cohomology!! ;-) We're just taking different resolutions of the appropriate constant sheaf.

IIRC, there are some good Cech cohomology computations and examples in Bott-Tu. Also, have you read section 3.H of Hatcher's algebraic topology book, on "local coefficients"?

For a simple example from algebraic geometry, compute the cohomology of the structure sheaf of $\mathbb{A}^2$ minus a point.

I seem to recall an exercise or an example in Hartshorne in which the genus of a degree $d$ curve in $\mathbb{P}^2$ is computed using Cech cohomology.

The section in Hartshorne on the cohomology of $\mathbb{P}^n$ uses Cech cohomology, and I remember finding it pretty instructive.

Eisenbud's commutative algebra book probably has lots of good examples.

Solution 2:

This is rather scheme-y, but there's a really nice paper by Kempf (hopefully you have institutional access :() that gives a very basic and elementary proof that the higher cohomology of a quasi-coherent sheaf on an affine scheme is trivial. The first part of the paper uses nothing more than the basic properties (e.g. long exact sequence) of cohomology, and might be fun. I thought it was fun, anyway; it's also nice because it shows that Hartshorne is unnecessarily restrictive in sticking to noetherian affine schemes in chapter III (even if one wants to avoid anything fancy).

OK, update: here is the proof explained (admittedly by a beginner :)).