Cohomology of twisting sheaves on projective space
= Cohomology of twisting sheaves on projective space
{title2=$H^q(\mathbb P^n,\mathcal O(d))$}
For every integer $d$, the intermediate cohomology of a twisting sheaf vanishes:
$$
H^q(\mathbb P_k^n,\mathcal O(d))=0\qquad(0<q<n).
$$
The zeroth cohomology for $d\geq0$ consists of degree-$d$ homogeneous polynomials, and the top cohomology is determined by <Serre duality>.