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The Stacks project

Lemma 71.2.6. 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 66.10.2).

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 67.15.4) 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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