The Stacks project

Proof. Let us denote $f : U \to S$ the structure morphism of $U$. By assumption (6) we can write $s^{-1}(\{ u\} ) = \{ r_1, \ldots , r_ n\} $. Since this set is finite, we see that $s$ is quasi-finite at each of these finitely many inverse images, see Morphisms, Lemma 29.20.7. Hence we see that $f \circ s : R \to S$ is quasi-finite at each $r_ i$ (Morphisms, Lemma 29.20.12). Hence $r_ i$ is isolated in the fibre $R_ p$, see Morphisms, Lemma 29.20.6. Write $t(\{ r_1, \ldots , r_ n\} ) = \{ u_1, \ldots , u_ m\} $. Note that it may happen that $m < n$ and note that $u \in \{ u_1, \ldots , u_ m\} $. Since $t$ is flat and locally of finite presentation, the morphism of fibres $t_ p : R_ p \to U_ p$ is flat and locally of finite presentation (Morphisms, Lemmas 29.25.8 and 29.21.4), hence open (Morphisms, Lemma 29.25.10). The fact that each $r_ i$ is isolated in $R_ p$ implies that each $u_ j = t(r_ i)$ is isolated in $U_ p$. Using Morphisms, Lemma 29.20.6 again, we see that $f$ is quasi-finite at $u_1, \ldots , u_ m$.

Denote $F_ u = s^{-1}(u)$ and $F_{u_ j} = s^{-1}(u_ j)$ the scheme theoretic fibres. Note that $F_ u$ is finite over $\kappa (u)$ as it is locally of finite type over $\kappa (u)$ with finitely many points (for example it follows from the much more general Morphisms, Lemma 29.57.9). By Lemma 40.7.1 we see that $F_ u$ and $F_{u_ j}$ become isomorphic over a common field extension of $\kappa (u)$ and $\kappa (u_ j)$. Hence we see that $F_{u_ j}$ is finite over $\kappa (u_ j)$. In particular we see $s^{-1}(\{ u_ j\} )$ is a finite set for each $j = 1, \ldots , m$. Thus we see that assumptions (2) and (6) hold for each $u_ j$ also (above we saw that $U \to S$ is quasi-finite at $u_ j$). Hence the argument of the first paragraph applies to each $u_ j$ and we see that $R \to U$ is quasi-finite at each of the points of

Note that $t(\{ r_1, \ldots , r_ N\} ) = \{ u_1, \ldots , u_ m\} $ and $t^{-1}(\{ u_1, \ldots , u_ m\} ) = \{ r_1, \ldots , r_ N\} $ since $R$ is a groupoid¹. Moreover, we have $\text{pr}_0(c^{-1}(\{ r_1, \ldots , r_ N\} )) = \{ r_1, \ldots , r_ N\} $ and $\text{pr}_1(c^{-1}(\{ r_1, \ldots , r_ N\} )) = \{ r_1, \ldots , r_ N\} $. Similarly we get $e(\{ u_1, \ldots , u_ m\} ) \subset \{ r_1, \ldots , r_ N\} $ and $i(\{ r_1, \ldots , r_ N\} ) = \{ r_1, \ldots , r_ N\} $.

We may apply More on Morphisms, Lemma 37.41.4 to the pairs $(U \to S, \{ u_1, \ldots , u_ m\} )$ and $(R \to S, \{ r_1, \ldots , r_ N\} )$ to get an étale neighbourhood $(S', p') \to (S, p)$ which induces an identification $\kappa (p) = \kappa (p')$ such that $S' \times _ S U$ and $S' \times _ S R$ decompose as

with $U' \to S'$ finite and $(U')_{p'}$ mapping bijectively to $\{ u_1, \ldots , u_ m\} $, and $R' \to S'$ finite and $(R')_{p'}$ mapping bijectively to $\{ r_1, \ldots , r_ N\} $. Moreover, no point of $W_{p'}$ (resp. $(W')_{p'}$) maps to any of the points $u_ j$ (resp. $r_ i$). At this point (a), (b), (c), and (d) of the lemma are satisfied. Moreover, the inclusions of (e) and (f) hold on fibres over $p'$, i.e., $s'((R')_{p'}) \subset (U')_{p'}$, $t'((R')_{p'}) \subset (U')_{p'}$, and $c'((R' \times _{s', U', t'} R')_{p'}) \subset (R')_{p'}$.

We claim that we can replace $S'$ by a Zariski open neighbourhood of $p'$ so that the inclusions of (e) and (f) hold. For example, consider the set $E = (s'|_{R'})^{-1}(W)$. This is open and closed in $R'$ and does not contain any points of $R'$ lying over $p'$. Since $R' \to S'$ is closed, after replacing $S'$ by $S' \setminus (R' \to S')(E)$ we reach a situation where $E$ is empty. In other words $s'$ maps $R'$ into $U'$. Note that this property is preserved under further shrinking $S'$. Similarly, we can arrange it so that $t'$ maps $R'$ into $U'$. At this point (e) holds. In the same manner, consider the set $E = (c'|_{R' \times _{s', U', t'} R'})^{-1}(W')$. It is open and closed in the scheme $R' \times _{s', U', t'} R'$ which is finite over $S'$, and does not contain any points lying over $p'$. Hence after replacing $S'$ by $S' \setminus (R' \times _{s', U', t'} R' \to S')(E)$ we reach a situation where $E$ is empty. In other words we obtain the inclusion in (f). We may repeat the argument also with the identity $e' : S' \times _ S U \to S' \times _ S R$ and the inverse $i' : S' \times _ S R \to S' \times _ S R$ so that we may assume (after shrinking $S'$ some more) that $(e'|_{U'})^{-1}(W') = \emptyset $ and $(i'|_{R'})^{-1}(W') = \emptyset $.

At this point we see that we may consider the structure

The axioms of a groupoid scheme over $S'$ hold because they hold for the groupoid scheme $(S' \times _ S U, S' \times _ S R, s', t', c', e', i')$. $\square$

Proof. We apply Lemma 40.13.1 for the groupoid $(U, R, s, t, c)$ over the scheme $S$ with points $p$ and $u$. Hence we get an étale neighbourhood $(S', p') \to (S, p)$ and disjoint union decompositions

and $u' \in U'$ satisfying conclusions (a), (b), (c), (d), (e), (f), and (g). We may shrink $S'$ to a smaller neighbourhood of $p'$ without affecting the conclusions (a) – (g). We will show that for a suitable shrinking conclusion (h) holds as well. Let us denote $j'$ the base change of $j$ to $S'$. By conclusion (e) it is clear that

for some open and closed $Rest$ piece. Since $U' \to S'$ is finite by conclusion (d) we see that $U' \times _{S'} U'$ is finite over $S'$. Since $j$ is universally closed, also $j'$ is universally closed, and hence $j'|_{Rest}$ is universally closed too. By conclusions (b) and (c) we see that the fibre of

over $p'$ is empty. Hence, since $Rest \to S'$ is closed as a composition of closed morphisms, after replacing $S'$ by $S' \setminus \mathop{\mathrm{Im}}(Rest \to S')$, we may assume that $Rest = \emptyset $. And this is exactly the condition that $R'$ is the restriction of $S' \times _ S R$ to the open subscheme $U' \subset S' \times _ S U$, see Groupoids, Lemma 39.18.3 and its proof. $\square$