= Cohomology of twisting sheaves on projective space
{title2=$H^q(\mathbb P^n,\mathcal O(d))$}
For every integer $d$, the cohomology of a <twisting sheaf on projective space> is
$$
H^q(\mathbb P_k^n,\mathcal O(d))\cong
\begin{cases}
k[x_0,\ldots,x_n]_d,&q=0\text{ and }d\geq0,\\
k[x_0,\ldots,x_n]_{-d-n-1}^*,&q=n\text{ and }d\leq-n-1,\\
0,&\text{otherwise}.
\end{cases}
$$
Thus all intermediate cohomology vanishes. The top line follows from <Serre duality> and $\omega_{\mathbb P^n}\cong\mathcal O(-n-1)$.
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