Hyperplane exact sequence for twisting sheaves ($0\to \mathcal{O}(m-1)\to \mathcal{O}(m)\to i_{\ast }{\mathcal{O}}_H(m)\to 0$)

For a hyperplane $H\subset {\mathbf{P}}^n$ with inclusion $i$, multiplication by its defining linear homogeneous polynomial gives the short exact sequence of sheavesThe 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 $n\ge 2$, where $h_r(m)=\dim H^r({\mathbf{P}}^r,\mathcal{O}(m))$.

The hyperplane exact sequence for twisting sheaves identifies consecutive top-degree groups whenever the hyperplane's preceding-degree group vanishes. Induction on dimension, starting with the surjective restriction of constants to a point on ${\mathbf{P}}^1$, propagates the degree-zero top vanishing down to $m=-n$. Its top-degree surjections also propagate vanishing to every positive twist. This proof does not apply the negative-twist global-section formula to ${\mathbf{P}}^0$.