= Ideal-sheaf vanishing for a coherent submodule of a trivial bundle
{title2=$\mathcal F\subseteq\mathcal O_X^r\Longrightarrow H^1(X,\mathcal F)=0$}
If $H^1(X,\mathcal I)=0$ for every <coherent ideal sheaf>, projection to the last coordinate makes any coherent $\mathcal F\subseteq\mathcal O_X^r$ an extension of a coherent submodule of $\mathcal O_X^{r-1}$ by an ideal sheaf. The <long exact sequence in sheaf cohomology> proves the assertion by <mathematical induction> on $r$.
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