The Stacks project

Lemma 115.11.3. 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. If $\mathcal{P}$ holds for a direct sum of coherent sheaves then it holds for both.

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

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

Proof. This follows from Lemma 115.11.2 in exactly the same way that Cohomology of Schemes, Lemma 30.12.6 follows from Cohomology of Schemes, Lemma 30.12.5. $\square$

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