# 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.

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No one uses injective resolutions to compute! – Mariano Suárez-Alvarez Aug 5 '10 at 6:02
Thank god! so what do people use in AG? And why have you not answered my MO question? ;) – Sean Tilson Aug 5 '10 at 6:17
In AG, people often use Cech computations, based on a finite cover by affine opens and the fact that higher cohomology of coherent sheaves vanishes on affine opens. They also use a result of Serre, to the effect that higher cohomology of any coherent sheaf twisted by a sufficiently positive line bundle vanishes. They use Riemann-Roch. In the analytic setting, they use the exponential short exact sequence, and in the etale cohomology setting, they use an etale analogue of this. They use the interpretation of $H^1$ of $\mathcal O^{\times}$ as the Picard group. They use Kodaira vanishing, – Matt E Aug 5 '10 at 6:44
when it applies, and other related vanishing theorems, they use Hodge theory (and especially Hodge symmetry, i.e. the equality $h^{p,q} = h^{q,p}$); of course, this list is not exhaustive, but it will give you some idea. Another very simple fact, but still often useful (especially when working on curves) is that skyscraper sheaves have vanishing higher cohomology. (More generally, if a sheaf is supported on some closed subspace $Y$ of $X$, we can compute its cohomology on $Y$ rather than $X$, which gives a lot of scope for inductive computations; this gives an interaction between the – Matt E Aug 5 '10 at 6:48
traditional approach, in projective geometry, of considering hyperplane sections, and the more modern viewpoint of using cohomology. For more, see Zariski's old (but beautiful) report on cohomology in algebraic geometry, from the Bulletin of the AMS in the 50s. – Matt E Aug 5 '10 at 6:49

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.

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Rotman does some very elementary explicit computations of Cech cohomology in his book Homological Algebra.

If I remember correctly he does these computations using resolutions, spectral sequences, and by just starting with some sequence.

As a complete beginner to this material, I was able to understand his treatment and compute some specific examples on my own.

I hope this helps :)

P.S. I just looked at the first edition, and it seems to be different slightly. For your information I used the second edition.

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What were your thoughts of the second edition as compared to the first, if any? – Sean Tilson Aug 5 '10 at 15:09
Supposedly he added/improved the second a lot. I don hav ethe first to compare. I should mention there are some typos in the second. I still recommend a look at this book despite. :) – BBischof Aug 5 '10 at 22:43
The relevant pages are 386-387 he does six examples, two are stupid, then he does reimann-roch. – BBischof Aug 5 '10 at 22:54

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 :)).

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In your quoted Theorem 1, "$latex{\mathcal{F}}&fg=000000$" did not get converted by whatever TeX renderer you use. – Larry Wang Aug 6 '10 at 22:32
Fixed, thanks for pointing it out. – Akhil Mathew Aug 7 '10 at 0:18