Lemma 40.14.5. Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ finite, $U$ is locally Noetherian, and $u_1, \ldots , u_ m \in U$ points whose orbits consist of generic points of irreducible components of $U$. Then there exist $R$-invariant subschemes $V' \subset V \subset U$ such that
$u_1, \ldots , u_ m \in V'$,
$V$ is open in $U$,
$V'$ and $V$ are affine,
$V' \subset V$ is a thickening,
the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$ are finite locally free.
Proof.
Let $\{ u_{j1}, \ldots , u_{jn_ j}\} $ be the orbit of $u_ j$. Let $W' \subset W \subset U$ be as in Lemma 40.14.2. Since $U = t(s^{-1}(\overline{W}))$ we see that at least one $u_{ji} \in \overline{W}$. Since $u_{ji}$ is a generic point of an irreducible component and $U$ locally Noetherian, this implies that $u_{ji} \in W$. Since $W$ is $R$-invariant, we conclude that $u_ j \in W$ and in fact the whole orbit is contained in $W$. By Cohomology of Schemes, Lemma 30.13.3 it suffices to find an $R$-invariant affine open subscheme $V'$ of $W'$ containing $u_1, \ldots , u_ m$ (because then we can let $V \subset W$ be the corresponding open subscheme which will be affine). Thus we may replace $(U, R, s, t, c)$ by the restriction $(W', R', s', t', c')$ to $W'$. In other words, we may assume we have a groupoid scheme $(U, R, s, t, c)$ whose morphisms $s$ and $t$ are finite locally free. By Properties, Lemma 28.29.1 we can find an affine open containing $\{ u_{ij}\} $ (a locally Noetherian scheme is quasi-separated by Properties, Lemma 28.5.4). Finally, we can apply Groupoids, Lemma 39.24.1 to conclude.
$\square$
Comments (0)