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40.13 Étale localization of groupoids

In this section we begin applying the étale localization techniques of More on Morphisms, Section 37.37 to groupoid schemes. More advanced material of this kind can be found in More on Groupoids in Spaces, Section 77.15. Lemma 40.13.2 will be used to prove results on algebraic spaces separated and quasi-finite over a scheme, namely Morphisms of Spaces, Proposition 65.50.2 and its corollary Morphisms of Spaces, Lemma 65.51.1.

Lemma 40.13.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $p \in S$ be a point, and let $u \in U$ be a point lying over $p$. Assume that

  1. $U \to S$ is locally of finite type,

  2. $U \to S$ is quasi-finite at $u$,

  3. $U \to S$ is separated,

  4. $R \to S$ is separated,

  5. $s$, $t$ are flat and locally of finite presentation, and

  6. $s^{-1}(\{ u\} )$ is finite.

Then there exists an étale neighbourhood $(S', p') \to (S, p)$ with $\kappa (p) = \kappa (p')$ and a base change diagram

\[ \xymatrix{ R' \amalg W' \ar@{=}[r] & S' \times _ S R \ar[r] \ar@<2ex>[d]^{s'} \ar@<-2ex>[d]_{t'} & R \ar@<1ex>[d]^ s \ar@<-1ex>[d]_ t \\ U' \amalg W \ar@{=}[r] & S' \times _ S U \ar[r] \ar[d] & U \ar[d] \\ & S' \ar[r] & S } \]

where the equal signs are decompositions into open and closed subschemes such that

  1. there exists a point $u'$ of $U'$ mapping to $u$ in $U$,

  2. the fibre $(U')_{p'}$ equals $t'\big ((s')^{-1}(\{ u'\} )\big )$ set theoretically,

  3. the fibre $(R')_{p'}$ equals $(s')^{-1}\big ((U')_{p'}\big )$ set theoretically,

  4. the schemes $U'$ and $R'$ are finite over $S'$,

  5. we have $s'(R') \subset U'$ and $t'(R') \subset U'$,

  6. we have $c'(R' \times _{s', U', t'} R') \subset R'$ where $c'$ is the base change of $c$, and

  7. the morphisms $s', t', c'$ determine a groupoid structure by taking the system $(U', R', s'|_{R'}, t'|_{R'}, c'|_{R' \times _{s', U', t'} R'})$.

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 29.56.9). 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

\[ \{ r_1, \ldots , r_ N\} = s^{-1}(\{ u_1, \ldots , u_ m\} ) \]

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 37.37.4 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

\[ S' \times _ S U = U' \amalg W, \quad S' \times _ S R = R' \amalg W' \]

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

\[ (U', R', s'|_{R'}, t'|_{R'}, c'|_{R' \times _{t', U', s'} R'}, e'|_{U'}, i'|_{R'}). \]

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

Lemma 40.13.2. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $p \in S$ be a point, and let $u \in U$ be a point lying over $p$. Assume assumptions (1) – (6) of Lemma 40.13.1 hold as well as

  1. $j : R \to U \times _ S U$ is universally closed2.

Then we can choose $(S', p') \to (S, p)$ and decompositions $S' \times _ S U = U' \amalg W$ and $S' \times _ S R = R' \amalg W'$ and $u' \in U'$ such that (a) – (g) of Lemma 40.13.1 hold as well as

  1. $R'$ is the restriction of $S' \times _ S R$ to $U'$.

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

\[ S' \times _ S U = U' \amalg W, \quad S' \times _ S R = R' \amalg W' \]

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

\[ j'^{-1}(U' \times _{S'} U') = R' \amalg Rest \]

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

\[ (U' \times _{S'} U' \to S') \circ j'|_{Rest} : Rest \longrightarrow S' \]

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. $\square$

[1] Explanation in groupoid language: The original set $\{ r_1, \ldots , r_ n\} $ was the set of arrows with source $u$. The set $\{ u_1, \ldots , u_ m\} $ was the set of objects isomorphic to $u$. And $\{ r_1, \ldots , r_ N\} $ is the set of all arrows between all the objects equivalent to $u$.
[2] In view of the other conditions this is equivalent to requiring $j$ to be proper.

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