Lemma 40.14.4. Let $(U, R, s, t, c)$ be a groupoid scheme over a scheme $S$. Assume $s, t$ are finite and of finite presentation and $U$ quasi-separated. Let $u_1, \ldots , u_ m \in U$ be 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

1. $u_1, \ldots , u_ m \in V'$,

2. $V$ is open in $U$,

3. $V'$ and $V$ are affine,

4. $V' \subset V$ is a thickening of finite presentation,

5. the morphisms $s', t'$ of the restriction $(V', R', s', t', c')$ are finite locally free.

Proof. Let $W' \subset W \subset U$ be as in Lemma 40.14.2. By Lemma 40.14.3 we get $u_ j \in W$ and that $W' \to W$ is a thickening of finite presentation. By Limits, Lemma 32.11.3 it suffices to find an $R$-invariant affine open subscheme $V'$ of $W'$ containing $u_ j$ (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 the union of the orbits of $u_1, \ldots , u_ m$. Finally, we can apply Groupoids, Lemma 39.24.1 to conclude. $\square$

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