The Stacks Project

Tag 03LO

Lemma 38.19.2. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.

1. For any subset $W \subset U$ the subset $t(s^{-1}(W))$ is set-theoretically $R$-invariant.
2. If $s$ and $t$ are open, then for every open $W \subset U$ the open $t(s^{-1}(W))$ is an $R$-invariant open subscheme.
3. If $s$ and $t$ are open and quasi-compact, then $U$ has an open covering consisting of $R$-invariant quasi-compact open subschemes.

Proof. Part (1) follows from Lemmas 38.3.4 and 38.13.2, namely, $t(s^{-1}(W))$ is the set of points of $U$ equivalent to a point of $W$. Next, assume $s$ and $t$ open and $W \subset U$ open. Since $s$ is open the set $W' = t(s^{-1}(W))$ is an open subset of $U$. Finally, assume that $s$, $t$ are both open and quasi-compact. Then, if $W \subset U$ is a quasi-compact open, then also $W' = t(s^{-1}(W))$ is a quasi-compact open, and invariant by the discussion above. Letting $W$ range over all affine opens of $U$ we see (3). $\square$

The code snippet corresponding to this tag is a part of the file groupoids.tex and is located in lines 3324–3336 (see updates for more information).

\begin{lemma}
\label{lemma-constructing-invariant-opens}
Let $S$ be a scheme.
Let $(U, R, s, t, c)$ be a groupoid scheme over $S$.
\begin{enumerate}
\item For any subset $W \subset U$ the subset $t(s^{-1}(W))$
is set-theoretically $R$-invariant.
\item If $s$ and $t$ are open, then for every open $W \subset U$
the open $t(s^{-1}(W))$ is an $R$-invariant open subscheme.
\item If $s$ and $t$ are open and quasi-compact, then $U$ has an open
covering consisting of $R$-invariant quasi-compact open subschemes.
\end{enumerate}
\end{lemma}

\begin{proof}
Part (1) follows from
Lemmas \ref{lemma-pre-equivalence-equivalence-relation-points} and
\ref{lemma-groupoid-pre-equivalence}, namely, $t(s^{-1}(W))$
is the set of points of $U$ equivalent to a point of $W$.
Next, assume $s$ and $t$ open and $W \subset U$ open.
Since $s$ is open the set $W' = t(s^{-1}(W))$ is an open subset of $U$.
Finally, assume that $s$, $t$ are both open and quasi-compact.
Then, if $W \subset U$ is a quasi-compact open, then also
$W' = t(s^{-1}(W))$ is a quasi-compact open, and invariant by the
discussion above. Letting $W$ range over all affine opens of $U$
we see (3).
\end{proof}

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