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 étale 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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)