The Stacks project

Lemma 70.2.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $x \in |X|$. If

  1. $x \in \text{Supp}(\mathcal{F})$

  2. $x$ is a codimension $0$ point of $X$ (Properties of Spaces, Definition 65.10.2).

Then $x \in \text{WeakAss}(\mathcal{F})$. If $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module with scheme theoretic support $Z$ (Morphisms of Spaces, Definition 66.15.4) and $x$ is a codimension $0$ point of $Z$, then $x \in \text{WeakAss}(\mathcal{F})$.

Proof. Since $x \in \text{Supp}(\mathcal{F})$ the stalk $\mathcal{F}_{\overline{x}}$ is not zero. Hence $\text{WeakAss}(\mathcal{F}_{\overline{x}})$ is nonempty by Algebra, Lemma 10.66.5. On the other hand, the spectrum of $\mathcal{O}_{X, \overline{x}}$ is a singleton. Hence $x$ is a weakly associated point of $\mathcal{F}$ by definition. The final statement follows as $\mathcal{O}_{X, \overline{x}} \to \mathcal{O}_{Z, \overline{z}}$ is a surjection, the spectrum of $\mathcal{O}_{Z, \overline{z}}$ is a singleton, and $\mathcal{F}_{\overline{x}}$ is a nonzero module over $\mathcal{O}_{Z, \overline{z}}$. $\square$


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