The Stacks project

Lemma 30.12.8. Let $X$ be a Noetherian scheme. Let $\mathcal{P}$ be a property of coherent sheaves on $X$ such that

  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. If $\mathcal{P}$ holds for $\mathcal{F}^{\oplus r}$ for some $r \geq 1$, then it holds for $\mathcal{F}$.

  3. For every integral closed subscheme $Z \subset X$ with generic point $\xi $ there exists some coherent sheaf $\mathcal{G}$ such that

    1. $\text{Supp}(\mathcal{G}) = Z$,

    2. $\mathcal{G}_\xi $ is annihilated by $\mathfrak m_\xi $, and

    3. for every quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_ X$ such that $\mathcal{J}_\xi = \mathcal{O}_{X, \xi }$ there exists a quasi-coherent subsheaf $\mathcal{G}' \subset \mathcal{J}\mathcal{G}$ with $\mathcal{G}'_\xi = \mathcal{G}_\xi $ and such that $\mathcal{P}$ holds for $\mathcal{G}'$.

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

Proof. Follows from Lemma 30.12.7 in exactly the same way that Lemma 30.12.6 follows from Lemma 30.12.5. $\square$

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