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Chapter 38: Groupoid Schemes > Section 38.19: Invariant subschemes

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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