The Stacks project

Lemma 20.55.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a sheaf of ideals. Consider the following two conditions

  1. for every $x \in X$ there exists an open neighbourhood $U \subset X$ of $x$ and $f \in \mathcal{I}(U)$ such that $\mathcal{I}|_ U = \mathcal{O}_ U \cdot f$ and $f : \mathcal{O}_ U \to \mathcal{O}_ U$ is injective, and

  2. $\mathcal{I}$ is invertible as an $\mathcal{O}_ X$-module.

Then (1) implies (2) and the converse is true if all stalks $\mathcal{O}_{X, x}$ of the structure sheaf are local rings.

Proof. Omitted. Hint: Use Modules, Lemma 17.25.4. $\square$

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