The Stacks project

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

  1. 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 $\mathcal{F}$.

  2. 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 $X$.

Proof. First note that if $\mathcal{F}$ is a coherent sheaf with a filtration

\[ 0 = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_ m = \mathcal{F} \]

by coherent subsheaves such that each of $\mathcal{F}_ i/\mathcal{F}_{i - 1}$ has property $\mathcal{P}$, then so does $\mathcal{F}$. This follows from the property (1) for $\mathcal{P}$. On the other hand, by Lemma 30.12.3 we can filter any $\mathcal{F}$ with successive subquotients as in (2). Hence the lemma follows. $\square$

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