Lemma 76.56.3. In Situation 76.56.1, assume $X$ is Noetherian. Then $X$ has the resolution property if and only if $\pi _*\mathcal{I}$ is the quotient of a finite locally free $\mathcal{O}_ X$-module.
Proof. The module $\pi _*\mathcal{I}$ is coherent by Cohomology of Spaces, Lemma 69.12.9. Hence if $X$ has the resolution property then $\pi _*\mathcal{I}$ is the quotient of a finite locally free $\mathcal{O}_ X$-module. Conversely, assume given a surjection $\mathcal{E} \to \pi _*\mathcal{I}$ for some finite locally free $\mathcal{O}_ X$-module $\mathcal{E}$. Observe that for all $n \geq 1$ there is a surjection
\[ \pi _*\mathcal{I} \otimes _{\mathcal{O}_ X} \pi _*\mathcal{I}^ n \longrightarrow \pi _*\mathcal{I}^{n + 1} \]
Hence $\mathcal{E}^{\otimes n}$ surjects onto $\pi _*\mathcal{I}^ n$ for all $n \geq 1$. We conclude that $X$ has the resolution property if we combine this with the result of Lemma 76.56.2. $\square$
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