The Stacks project

Lemma 30.12.6. 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 if two out of three of them have property $\mathcal{P}$ then so does the third.

  2. 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 $,

    3. $\dim _{\kappa (\xi )} \mathcal{G}_\xi = 1$, and

    4. property $\mathcal{P}$ holds for $\mathcal{G}$.

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

Proof. According to Lemma 30.12.4 it suffices to show that for all integral closed subschemes $Z \subset X$ and all quasi-coherent ideal sheaves $\mathcal{I} \subset \mathcal{O}_ Z$ we have $\mathcal{P}$ for $(Z \to X)_*\mathcal{I}$. If this fails, then since $X$ is Noetherian there is a minimal integral closed subscheme $Z_0 \subset X$ such that $\mathcal{P}$ fails for $(Z_0 \to X)_*\mathcal{I}_0$ for some quasi-coherent sheaf of ideals $\mathcal{I}_0 \subset \mathcal{O}_{Z_0}$, but $\mathcal{P}$ does hold for $(Z \to X)_*\mathcal{I}$ for all integral closed subschemes $Z \subset Z_0$, $Z \not= Z_0$ and quasi-coherent ideal sheaves $\mathcal{I} \subset \mathcal{O}_ Z$. Since we have the existence of $\mathcal{G}$ for $Z_0$ by part (2), according to Lemma 30.12.5 this cannot happen. $\square$

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