Lemma 30.12.4. Let $X$ be a Noetherian scheme. Let $\mathcal{P}$ be a property of coherent sheaves on $X$. Assume

For any short exact sequence of coherent sheaves

\[ 0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0 \]if $\mathcal{F}_ i$, $i = 1, 2$ have property $\mathcal{P}$ then so does $\mathcal{F}$.

For every integral closed subscheme $Z \subset X$ and every quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ Z$ we have $\mathcal{P}$ for $i_*\mathcal{I}$.

Then property $\mathcal{P}$ holds for every coherent sheaf on $X$.

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