Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 16 / 2 / a / Solution

For $r=1$, the assertion is the assumed vanishing for coherent ideal sheaves. For $r>1$, project $\mathcal{F}\subseteq {\mathcal{O}}_X^r$ onto the last component. Its image $\mathcal{I}\subseteq {\mathcal{O}}_X$ is a coherent ideal sheaf, and its kernel ${\mathcal{F}}^{\prime }$ is a coherent sheaf contained in ${\mathcal{O}}_X^{r-1}$. Here images and kernels are coherent because a variety is Noetherian. The short exact sequencegives an exact segment $H^1(X,{\mathcal{F}}^{\prime })\to H^1(X,\mathcal{F})\to H^1(X,\mathcal{I})$ in the long exact sequence in sheaf cohomology. The outer terms vanish by mathematical induction and the hypothesis, so the middle term vanishes. This is ideal-sheaf vanishing for a coherent submodule of a trivial bundle.