Identifying sections when computing cohomology groups $H^i(\mathbb P^2,\mathcal O(d))$ via Cech cohomology

Question

I want to compute the dimension and find generators of the cohomology groups $H^i(\mathbb P^2,\mathcal O(d))$. To do so I want to use Cech cohomology and compute directly, or at least try to. So we have a cover with affine spaces and we will get something like $0\to \oplus_{i=0}^2\Gamma(U_i,\mathcal O(d))\to \oplus_{i<j}(\Gamma(U_i\cap U_j,\mathcal O(d)))\to \Gamma(U_0\cap U_1\cap U_2,\mathcal O(d))$.

However I've always struggled since I saw these sheaves, I get their definition but I don't know how to interpret them. Here for example, what would the sections be? For the first direct sum I would suggest the homogenous polynomials of degree $d$.

I know that there is a theorem that takes care of this but I think that if I knew what was going on in this little example I would understand better how to compute cohomology groups with Cech.

Answer 1

The key here is the following statement:

Let $S$ be a graded ring, let $f\in S$ be homogeneous of positive degree, and let $M$ be a graded $S$-module. Then the sections of $\widetilde{M}$ over the affine open set $D_+(f)\subset\operatorname{Proj} S$ are exactly $M_{(f)}=(M_f)_0$, the homogeneous localization of the module $M$ at $f$.

This is Hartshorne proposition II.5.11 or Stacks 01M7, for instance.

To apply this, note that your $U_i$ is $D_+(x_i)$ and $\mathcal{O}(d)$ is $\widetilde{k[x_0,\cdots,x_n](d)}$, so the sections over $U_i$ are $x_i^dk[\frac{x_0}{x_i},\cdots,\frac{x_n}{x_0}]$. A similar argument, recognizing $U_i\cap U_j=D_+(x_ix_j)$ tells you what the sections are over the double and triple intersections, respectively.

If you're having other issues with the proof, you may want to look at how it's done in a few different sources - there's essentially one main method of proof, but different authors can provide different levels of detail at different steps. I know Vakil, Hartshorne, and Stacks' expositions are all slightly different.