Cohomology of twisting sheaves on projective space ($H^q({\mathbb{P}}^n,\mathcal{O}(d))$)

For every integer $d$, the cohomology of a twisting sheaf on projective space isThus all intermediate cohomology vanishes. The top line follows from Serre duality and ${\omega }_{{\mathbb{P}}^n}\cong \mathcal{O}(-n-1)$.

For $m\ge 0$, these sheaf cohomology dimensions follow from the vanishing for $\mathcal{O}$ and the hyperplane exact sequence for twisting sheaves. Induct on dimension and then on $m$ in $0\to \mathcal{O}(m-1)\to \mathcal{O}(m)\to i_{\ast }{\mathcal{O}}_{{\mathbf{P}}^{n-1}}(m)\to 0$. The long exact sequence in sheaf cohomology gives higher-degree vanishing and the recurrence $h_n^0(m)-h_n^0(m-1)=\left(\genfrac{}{}{0ex}{}{n-1+m}{n-1}\right)$. Starting with $h_n^0(0)=1$ yields the formula; the corresponding homogeneous monomials form a basis of global sections.

On the projective line, an infinite direct sum of copies of $\mathcal{O}(-1)$ is a quasi-coherent sheaf but not a coherent sheaf: at any point its residue-vector-space dimension is infinite. Its global sections vanish because each summand has no sections, and sections commute with direct sums for the finite two-chart equalizer. Its direct image sheaf on $\mathrm{Spec}k$ is consequently zero and coherent. This shows that a nonaffine direct image can lose information needed to detect coherence.

The only regular functions defined on all of a nonempty projective space over a field $k$ are the constants:For ${\mathbb{P}}^1$, this also follows by gluing $k[t]$ and $k[t^{-1}]$ inside $k[t,t^{-1}]$: their intersection is $k$.