As a consequence of the vanishing of cohomology for affine schemes: for a separated scheme X, an affine open covering {Ui} of X, and a quasi-coherent sheaf E on X, the cohomology groups H*(X,E) are isomorphic to the Čech cohomology groups with respect to the open covering {Ui} [REF].
In particular, this calculation shows that the cohomology of projective space over k with coefficients in any line bundle has finite dimension as a k-vector space.