# The Stacks Project

## Tag 0APA

Lemma 38.19.3. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume $s$ and $t$ quasi-compact and flat and $U$ quasi-separated. Let $W \subset U$ be quasi-compact open. Then $t(s^{-1}(W))$ is an intersection of a nonempty family of quasi-compact open subsets of $U$.

Proof. Note that $s^{-1}(W)$ is quasi-compact open in $R$. As a continuous map $t$ maps the quasi-compact subset $s^{-1}(W)$ to a quasi-compact subset $t(s^{-1}(W))$. As $t$ is flat and $s^{-1}(W)$ is closed under generalization, so is $t(s^{-1}(W))$, see (Morphisms, Lemma 28.24.8 and Topology, Lemma 5.19.5). Pick a quasi-compact open $W' \subset U$ containing $t(s^{-1}(W))$. By Properties, Lemma 27.2.4 we see that $W'$ is a spectral space (here we use that $U$ is quasi-separated). Then the lemma follows from Topology, Lemma 5.24.7 applied to $t(s^{-1}(W)) \subset W'$. $\square$

The code snippet corresponding to this tag is a part of the file groupoids.tex and is located in lines 3352–3358 (see updates for more information).

\begin{lemma}
\label{lemma-first-observation}
Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
Assume $s$ and $t$ quasi-compact and flat and $U$ quasi-separated.
Let $W \subset U$ be quasi-compact open. Then $t(s^{-1}(W))$
is an intersection of a nonempty family of quasi-compact open subsets of $U$.
\end{lemma}

\begin{proof}
Note that $s^{-1}(W)$ is quasi-compact open in $R$.
As a continuous map $t$ maps the quasi-compact subset
$s^{-1}(W)$ to a quasi-compact subset $t(s^{-1}(W))$.
As $t$ is flat and $s^{-1}(W)$ is closed under generalization,
so is $t(s^{-1}(W))$, see
(Morphisms, Lemma \ref{morphisms-lemma-generalizations-lift-flat} and
Topology, Lemma \ref{topology-lemma-lift-specializations-images}).
Pick a quasi-compact open $W' \subset U$ containing $t(s^{-1}(W))$. By
Properties, Lemma \ref{properties-lemma-quasi-compact-quasi-separated-spectral}
we see that $W'$ is a spectral space (here we use that $U$ is quasi-separated).
Then the lemma follows from
Topology, Lemma \ref{topology-lemma-make-spectral-space}
applied to $t(s^{-1}(W)) \subset W'$.
\end{proof}

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