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The Stacks project

Lemma 71.2.7. 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 is decent (for example quasi-separated or locally separated),

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

  3. x is not a specialization of another point in \text{Supp}(\mathcal{F}).

Then x \in \text{WeakAss}(\mathcal{F}).

Proof. (A quasi-separated algebraic space is decent, see Decent Spaces, Section 68.6. A locally separated algebraic space is decent, see Decent Spaces, Lemma 68.15.2.) Choose a scheme U, a point u \in U, and an étale morphism f : U \to X mapping u to x. By Decent Spaces, Lemma 68.12.1 if u' \leadsto u is a nontrivial specialization, then f(u') \not= x. Hence we see that u \in \text{Supp}(f^*\mathcal{F}) is not a specialization of another point of \text{Supp}(f^*\mathcal{F}). Hence u \in \text{WeakAss}(f^*\mathcal{F}) by Divisors, Lemma 71.2.6. \square


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