Lemma 76.23.5. Let $S$ be a scheme. Let $f : X \to Y$ and $Y \to Z$ be morphisms of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume

1. $X$ is locally of finite presentation over $Z$,

2. $\mathcal{F}$ an $\mathcal{O}_ X$-module of finite presentation,

3. $\mathcal{F}$ is flat over $Z$, and

4. $Y$ is locally of finite type over $Z$.

Then the set

$A = \{ x \in |X| : \mathcal{F} \text{ flat at }x \text{ over }Y\} .$

is open in $|X|$ and its formation commutes with arbitrary base change: If $Z' \to Z$ is a morphism of algebraic spaces, and $A'$ is the set of points of $X' = X \times _ Z Z'$ where $\mathcal{F}' = \mathcal{F} \times _ Z Z'$ is flat over $Y' = Y \times _ Z Z'$, then $A'$ is the inverse image of $A$ under the continuous map $|X'| \to |X|$.

Proof. One way to prove this is to translate the proof as given in More on Morphisms, Lemma 37.16.4 into the category of algebraic spaces. Instead we will prove this by reducing to the case of schemes. Namely, choose a diagram as in Lemma 76.23.1 part (3) such that $a$, $b$, and $c$ are surjective. It follows from the definitions that this reduces to the corresponding theorem for the morphisms of schemes $U \to V \to W$, the quasi-coherent sheaf $a^*\mathcal{F}$, and the point $u \in U$. The only minor point to make is that given a morphism of algebraic spaces $Z' \to Z$ we choose a scheme $W'$ and a surjective étale morphism $W' \to W \times _ Z Z'$. Then we set $U' = W' \times _ W U$ and $V' = W' \times _ W V$. We write $a', b', c'$ for the morphisms from $U', V', W'$ to $X', Y', Z'$. In this case $A$, resp. $A'$ are images of the open subsets of $U$, resp. $U'$ associated to $a^*\mathcal{F}$, resp. $(a')^*\mathcal{F}'$. This indeed does reduce the lemma to More on Morphisms, Lemma 37.16.4. $\square$

## 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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05X2. Beware of the difference between the letter 'O' and the digit '0'.