The Stacks project

Lemma 20.53.3. In Situation 20.53.2 let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. The following are equivalent

  1. the subsheaf $\mathcal{F}[\mathcal{I}] \subset \mathcal{F}$ of sections annihilated by $\mathcal{I}$ is zero,

  2. the subsheaf $\mathcal{F}[\mathcal{I}^ n]$ is zero for all $n \geq 1$,

  3. the multiplication map $\mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{F}$ is injective,

  4. for every open $U \subset X$ such that $\mathcal{I}|_ U = \mathcal{O}_ U \cdot f$ for some $f \in \mathcal{I}(U)$ the map $f : \mathcal{F}|_ U \to \mathcal{F}|_ U$ is injective,

  5. for every $x \in X$ and generator $f$ of the ideal $\mathcal{I}_ x \subset \mathcal{O}_{X, x}$ the element $f$ is a nonzerodivisor on the stalk $\mathcal{F}_ x$.

Proof. Omitted. $\square$


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