The Stacks project

Lemma 76.6.1. Let $S$ be a scheme. Let $X$ be a decent algebraic space locally of finite type over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ X$-module. Let $s \in S$ such that $\mathcal{F}$ is flat over $S$ at all points of $X_ s$. Let $x' \in \text{Ass}_{X/S}(\mathcal{F})$. If the closure of $\{ x'\} $ in $|X|$ meets $|X_ s|$, then the closure meets $\text{Ass}_{X/S}(\mathcal{F}) \cap |X_ s|$.

Proof. Observe that $|X_ s| \subset |X|$ is the set of points of $|X|$ lying over $s \in S$, see Decent Spaces, Lemma 67.18.6. Let $t \in |X_ s|$ be a specialization of $x'$ in $|X|$. Choose an affine scheme $U$ and a point $u \in U$ and an ├ętale morphism $\varphi : U \to X$ mapping $u$ to $t$. By Decent Spaces, Lemma 67.12.2 we can choose a specialization $u' \leadsto u$ with $u'$ mapping to $x'$. Set $g = f \circ \varphi $. Observe that $s' = g(u') = f(x')$ specializes to $s$. By our definition of $\text{Ass}_{X/S}(\mathcal{F})$ we have $u' \in \text{Ass}_{U/S}(\varphi ^*\mathcal{F})$. By the schemes version of this lemma (More on Flatness, Lemma 38.18.1) we see that there is a specialization $u' \leadsto u$ with $u \in \text{Ass}_{U_ s}(\varphi ^*\mathcal{F}_ s) = \text{Ass}_{U/S}(\varphi ^*\mathcal{F}) \cap U_ s$. Hence $x = \varphi (u) \in \text{Ass}_{X/S}(\mathcal{F})$ lies over $s$ and the lemma is proved. $\square$


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