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 $X$.