TITLE: Why is $H^n(\mathbb{P}_k^n, \omega) \cong k$?
QUESTION [7 upvotes]: Hartshorne makes this sound like a coincidence: we us Cech cohomology on the usual open affine cover $\mathcal{U}$ to get the chain complex
$$\check{C}^\bullet\left(\mathcal{U}, \bigoplus_{n \in \mathbb{Z}} \mathcal{O}(n)\right): 0 \to \bigoplus_i S_{x_i} \to \cdots \to \bigoplus_i S_{x_0\cdots\hat{x_i}\cdots x_r} \to S_{x_0 \cdots x_r}\to 0,$$
calculate the cokernel of the last map to obtain the cohomology groups $\check{H}^r(\mathcal{U}, \mathcal{O}(n))$, and then observe that it's one dimensional when $n = -r - 1$.  Voilà!  What a coincidence!
Surely there's an actual reason here, though.  What is it?

REPLY [6 votes]: This is a very long and interesting story.
In a nutshell: it is Serre duality, which asserts that for a smooth projective variety $X$ of dimension $n$ and a locally free sheaf $V$  on $X$ we have  a canonical isomorphism $$  H^i(X, V)\cong (H^{n-i}(X, \check{V}\otimes \omega))^*                $$ [The Čech sign over $V$ denotes dual vector bundle and the star denotes dual $k$-vector space ]     
Your case follows by taking $X=\mathbb P^n, V=\omega, i=n$ and noticing that $\omega=\mathcal O_{\mathbb P^n}(-n-1)$.  
Edit
Serre introduced his duality theorem in  "Un théorème de dualité" , Comm.Math.Helv. 29 (1955) Théorème 4.
He worked in the context of holomorphic manifolds and the cohomology he uses is Dolbeault cohomology.
The algebraic analogue was developed by Grothendieck and his school.
The cohomology they use is  derived functor cohomology, as introduced  by Grothendieck in his Tohoku paper.