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 ├ętale 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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