= Hyperplane exact sequence for twisting sheaves
{title2=$0\to\mathcal O(m-1)\to\mathcal O(m)\to i_*\mathcal O_H(m)\to0$}
For a hyperplane $H\subset\mathbf P^n$ with inclusion $i$, multiplication by its defining linear <homogeneous polynomial> gives the <short exact sequence of sheaves>
$$
0\to\mathcal O_{\mathbf P^n}(m-1)\to\mathcal O_{\mathbf P^n}(m)\to i_*\mathcal O_H(m)\to0.
$$
The <long exact sequence in sheaf cohomology> relates twists in consecutive degrees and consecutive dimensions. Surjectivity of restriction on global homogeneous polynomials, together with eventual positive-twist vanishing, proves vanishing of intermediate cohomology by descending induction. The top-degree dimensions satisfy $h_n(m-1)=h_n(m)+h_{n-1}(m)$ for $n\geq2$, where $h_r(m)=\dim H^r(\mathbf P^r,\mathcal O(m))$.
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