79.15 Étale localization of groupoid schemes
In this section we prove results similar to [Proposition 4.2, K-M]. We try to be a bit more general, and we try to avoid using Hilbert schemes by using the finite part of a morphism instead. The goal is to "split" a groupoid in algebraic spaces over a point after étale localization. Here is the definition (very similar to [Definition 4.1, K-M]).
Definition 79.15.1. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$ Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $u \in |U|$ be a point.
We say $R$ is strongly split over $u$ if there exists an open subspace $P \subset R$ such that
$(U, P, s|_ P, t|_ P, c|_{P \times _{s, U, t} P})$ is a groupoid in algebraic spaces over $B$,
$s|_ P$, $t|_ P$ are finite, and
$\{ r \in |R| : s(r) = u, t(r) = u\} \subset |P|$.
The choice of such a $P$ will be called a strong splitting of $R$ over $u$.
We say $R$ is split over $u$ if there exists an open subspace $P \subset R$ such that
$(U, P, s|_ P, t|_ P, c|_{P \times _{s, U, t} P})$ is a groupoid in algebraic spaces over $B$,
$s|_ P$, $t|_ P$ are finite, and
$\{ g \in |G| : g\text{ maps to }u\} \subset |P|$ where $G \to U$ is the stabilizer.
The choice of such a $P$ will be called a splitting of $R$ over $u$.
We say $R$ is quasi-split over $u$ if there exists an open subspace $P \subset R$ such that
$(U, P, s|_ P, t|_ P, c|_{P \times _{s, U, t} P})$ is a groupoid in algebraic spaces over $B$,
$s|_ P$, $t|_ P$ are finite, and
$e(u) \in |P|$1.
The choice of such a $P$ will be called a quasi-splitting of $R$ over $u$.
Note the similarity of the conditions on $P$ to the conditions on pairs in (79.12.0.1). In particular, if $s, t$ are separated, then $P$ is also closed in $R$ (see Lemma 79.12.4).
Suppose we start with a groupoid in algebraic spaces $(U, R, s, t, c)$ over $B$ and a point $u \in |U|$. Since the goal is to split the groupoid after étale localization we may as well replace $U$ by an affine scheme (what we mean is that this is harmless for any possible application). Moreover, the additional hypotheses we are going to have to impose will force $R$ to be a scheme at least in a neighbourhood of $\{ r \in |R| : s(r) = u, t(r) = u\} $ or $e(u)$. This is why we start with a groupoid scheme as described below. However, our technique of proof leads us outside of the category of schemes, which is why we have formulated a splitting for the case of groupoids in algebraic spaces above. On the other hand, we know of no applications but the case where the morphisms $s$, $t$ are also flat and of finite presentation, in which case we end up back in the category of schemes.
Situation 79.15.2 (Strong splitting). Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $u \in U$ be a point. Assume that
$s, t : R \to U$ are separated,
$s$, $t$ are locally of finite type,
the set $\{ r \in R : s(r) = u, t(r) = u\} $ is finite, and
$s$ is quasi-finite at each point of the set in (3).
Note that assumptions (3) and (4) are implied by the assumption that the fibre $s^{-1}(\{ u\} )$ is finite, see Morphisms, Lemma 29.20.7.
Situation 79.15.3 (Splitting). Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $u \in U$ be a point. Assume that
$s, t : R \to U$ are separated,
$s$, $t$ are locally of finite type,
the set $\{ g \in G : g\text{ maps to }u\} $ is finite where $G \to U$ is the stabilizer, and
$s$ is quasi-finite at each point of the set in (3).
Situation 79.15.4 (Quasi-splitting). Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $u \in U$ be a point. Assume that
$s, t : R \to U$ are separated,
$s$, $t$ are locally of finite type, and
$s$ is quasi-finite at $e(u)$.
For our application to the existence theorems for algebraic spaces the case of quasi-splittings is sufficient. Moreover, the quasi-splitting case will allow us to prove an étale local structure theorem for quasi-DM stacks. The splitting case will be used to prove a version of the Keel-Mori theorem. The strong splitting case applies to give an étale local structure theorem for quasi-DM algebraic stacks with quasi-compact diagonal.
Lemma 79.15.5 (Existence of strong splitting). In Situation 79.15.2 there exists an algebraic space $U'$, an étale morphism $U' \to U$, and a point $u' : \mathop{\mathrm{Spec}}(\kappa (u)) \to U'$ lying over $u : \mathop{\mathrm{Spec}}(\kappa (u)) \to U$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is strongly split over $u'$.
Proof.
Let $f : (U', Z_{univ}, s', t', c') \to (U, R, s, t, c)$ be as constructed in Lemma 79.14.1. Recall that $R' = R \times _{(U \times _ S U)} (U' \times _ S U')$. Thus we get a morphism $(f, t', s') : Z_{univ} \to R'$ of groupoids in algebraic spaces
\[ (U', Z_{univ}, s', t', c') \to (U', R', s', t', c') \]
(by abuse of notation we indicate the morphisms in the two groupoids by the same symbols). Now, as $Z_{univ} \subset R \times _{s, U, g} U'$ is open and $R' \to R \times _{s, U, g} U'$ is étale (as a base change of $U' \to U$) we see that $Z_{univ} \to R'$ is an open immersion. By construction the morphisms $s', t' : Z_{univ} \to U'$ are finite. It remains to find the point $u'$ of $U'$.
We think of $u$ as a morphism $\mathop{\mathrm{Spec}}(\kappa (u)) \to U$ as in the statement of the lemma. Set $F_ u = R \times _{s, U} \mathop{\mathrm{Spec}}(\kappa (u))$. The set $\{ r \in R : s(r) = u, t(r) = u\} $ is finite by assumption and $F_ u \to \mathop{\mathrm{Spec}}(\kappa (u))$ is quasi-finite at each of its elements by assumption. Hence we can find a decomposition into open and closed subschemes
\[ F_ u = Z_ u \amalg Rest \]
for some scheme $Z_ u$ finite over $\kappa (u)$ whose support is $\{ r \in R : s(r) = u, t(r) = u\} $. Note that $e(u) \in Z_ u$. Hence by the construction of $U'$ in Section 79.14 $(u, Z_ u)$ defines a $\mathop{\mathrm{Spec}}(\kappa (u))$-valued point $u'$ of $U'$.
We still have to show that the set $\{ r' \in |R'| : s'(r') = u', t'(r') = u'\} $ is contained in $|Z_{univ}|$. Pick any point $r'$ in this set and represent it by a morphism $z' : \mathop{\mathrm{Spec}}(k) \to R'$. Denote $z : \mathop{\mathrm{Spec}}(k) \to R$ the composition of $z'$ with the map $R' \to R$. Clearly, $z$ defines an element of the set $\{ r \in R : s(r) = u, t(r) = u\} $. Also, the compositions $s \circ z, t \circ z : \mathop{\mathrm{Spec}}(k) \to U$ factor through $u$, so we may think of $s \circ z, t \circ z$ as a morphism $\mathop{\mathrm{Spec}}(k) \to \mathop{\mathrm{Spec}}(\kappa (u))$. Then $z' = (z, u' \circ t \circ z, u'\circ s \circ u)$ as morphisms into $R' = R \times _{(U \times _ S U)} (U' \times _ S U')$. Consider the triple
\[ (s \circ z, Z_ u \times _{\mathop{\mathrm{Spec}}(\kappa (u)), s \circ z} \mathop{\mathrm{Spec}}(k), z) \]
where $Z_ u$ is as above. This defines a $\mathop{\mathrm{Spec}}(k)$-valued point of $Z_{univ}$ whose image via $s', t'$ in $U'$ is $u'$ and whose image via $Z_{univ} \to R'$ is the point $r'$ by the relationship between $z$ and $z'$ mentioned above. This finishes the proof.
$\square$
Lemma 79.15.6 (Existence of splitting). In Situation 79.15.3 there exists an algebraic space $U'$, an étale morphism $U' \to U$, and a point $u' : \mathop{\mathrm{Spec}}(\kappa (u)) \to U'$ lying over $u : \mathop{\mathrm{Spec}}(\kappa (u)) \to U$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is split over $u'$.
Proof.
Let $f : (U', Z_{univ}, s', t', c') \to (U, R, s, t, c)$ be as constructed in Lemma 79.14.1. Recall that $R' = R \times _{(U \times _ S U)} (U' \times _ S U')$. Thus we get a morphism $(f, t', s') : Z_{univ} \to R'$ of groupoids in algebraic spaces
\[ (U', Z_{univ}, s', t', c') \to (U', R', s', t', c') \]
(by abuse of notation we indicate the morphisms in the two groupoids by the same symbols). Now, as $Z_{univ} \subset R \times _{s, U, g} U'$ is open and $R' \to R \times _{s, U, g} U'$ is étale (as a base change of $U' \to U$) we see that $Z_{univ} \to R'$ is an open immersion. By construction the morphisms $s', t' : Z_{univ} \to U'$ are finite. It remains to find the point $u'$ of $U'$.
We think of $u$ as a morphism $\mathop{\mathrm{Spec}}(\kappa (u)) \to U$ as in the statement of the lemma. Set $F_ u = R \times _{s, U} \mathop{\mathrm{Spec}}(\kappa (u))$. Let $G_ u \subset F_ u$ be the scheme theoretic fibre of $G \to U$ over $u$. By assumption $G_ u$ is finite and $F_ u \to \mathop{\mathrm{Spec}}(\kappa (u))$ is quasi-finite at each point of $G_ u$ by assumption. Hence we can find a decomposition into open and closed subschemes
\[ F_ u = Z_ u \amalg Rest \]
for some scheme $Z_ u$ finite over $\kappa (u)$ whose support is $G_ u$. Note that $e(u) \in Z_ u$. Hence by the construction of $U'$ in Section 79.14 $(u, Z_ u)$ defines a $\mathop{\mathrm{Spec}}(\kappa (u))$-valued point $u'$ of $U'$.
We still have to show that the set $\{ g' \in |G'| : g'\text{ maps to }u'\} $ is contained in $|Z_{univ}|$. Pick any point $g'$ in this set and represent it by a morphism $z' : \mathop{\mathrm{Spec}}(k) \to G'$. Denote $z : \mathop{\mathrm{Spec}}(k) \to G$ the composition of $z'$ with the map $G' \to G$. Clearly, $z$ defines a point of $G_ u$. In fact, let us write $\tilde u : \mathop{\mathrm{Spec}}(k) \to u \to U$ for the corresponding map to $u$ or $U$. Consider the triple
\[ (\tilde u, Z_ u \times _{u, \tilde u} \mathop{\mathrm{Spec}}(k), z) \]
where $Z_ u$ is as above. This defines a $\mathop{\mathrm{Spec}}(k)$-valued point of $Z_{univ}$ whose image via $s', t'$ in $U'$ is $u'$ and whose image via $Z_{univ} \to R'$ is the point $z'$ (because the image in $R$ is $z$). This finishes the proof.
$\square$
Lemma 79.15.7 (Existence of quasi-splitting). In Situation 79.15.4 there exists an algebraic space $U'$, an étale morphism $U' \to U$, and a point $u' : \mathop{\mathrm{Spec}}(\kappa (u)) \to U'$ lying over $u : \mathop{\mathrm{Spec}}(\kappa (u)) \to U$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is quasi-split over $u'$.
Proof.
Let $f : (U', Z_{univ}, s', t', c') \to (U, R, s, t, c)$ be as constructed in Lemma 79.14.1. Recall that $R' = R \times _{(U \times _ S U)} (U' \times _ S U')$. Thus we get a morphism $(f, t', s') : Z_{univ} \to R'$ of groupoids in algebraic spaces
\[ (U', Z_{univ}, s', t', c') \to (U', R', s', t', c') \]
(by abuse of notation we indicate the morphisms in the two groupoids by the same symbols). Now, as $Z_{univ} \subset R \times _{s, U, g} U'$ is open and $R' \to R \times _{s, U, g} U'$ is étale (as a base change of $U' \to U$) we see that $Z_{univ} \to R'$ is an open immersion. By construction the morphisms $s', t' : Z_{univ} \to U'$ are finite. It remains to find the point $u'$ of $U'$.
We think of $u$ as a morphism $\mathop{\mathrm{Spec}}(\kappa (u)) \to U$ as in the statement of the lemma. Set $F_ u = R \times _{s, U} \mathop{\mathrm{Spec}}(\kappa (u))$. The morphism $F_ u \to \mathop{\mathrm{Spec}}(\kappa (u))$ is quasi-finite at $e(u)$ by assumption. Hence we can find a decomposition into open and closed subschemes
\[ F_ u = Z_ u \amalg Rest \]
for some scheme $Z_ u$ finite over $\kappa (u)$ whose support is $e(u)$. Hence by the construction of $U'$ in Section 79.14 $(u, Z_ u)$ defines a $\mathop{\mathrm{Spec}}(\kappa (u))$-valued point $u'$ of $U'$. To finish the proof we have to show that $e'(u') \in Z_{univ}$ which is clear.
$\square$
Finally, when we add additional assumptions we obtain schemes.
Lemma 79.15.8. In Situation 79.15.2 assume in addition that $s, t$ are flat and locally of finite presentation. Then there exists a scheme $U'$, a separated étale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa (u) = \kappa (u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is strongly split over $u'$.
Proof.
This follows from the construction of $U'$ in the proof of Lemma 79.15.5 because in this case $U' = (R_ s/U, e)_{fin}$ is a scheme separated over $U$ by Lemmas 79.12.14 and 79.12.15.
$\square$
Lemma 79.15.9. In Situation 79.15.3 assume in addition that $s, t$ are flat and locally of finite presentation. Then there exists a scheme $U'$, a separated étale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa (u) = \kappa (u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is split over $u'$.
Proof.
This follows from the construction of $U'$ in the proof of Lemma 79.15.6 because in this case $U' = (R_ s/U, e)_{fin}$ is a scheme separated over $U$ by Lemmas 79.12.14 and 79.12.15.
$\square$
Lemma 79.15.10. In Situation 79.15.4 assume in addition that $s, t$ are flat and locally of finite presentation. Then there exists a scheme $U'$, a separated étale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa (u) = \kappa (u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is quasi-split over $u'$.
Proof.
This follows from the construction of $U'$ in the proof of Lemma 79.15.7 because in this case $U' = (R_ s/U, e)_{fin}$ is a scheme separated over $U$ by Lemmas 79.12.14 and 79.12.15.
$\square$
In fact we can obtain affine schemes by applying an earlier result on finite locally free groupoids.
Lemma 79.15.11. In Situation 79.15.2 assume in addition that $s, t$ are flat and locally of finite presentation and that $U$ is affine. Then there exists an affine scheme $U'$, an étale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa (u) = \kappa (u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is strongly split over $u'$.
Proof.
Let $U' \to U$ and $u' \in U'$ be the separated étale morphism of schemes we found in Lemma 79.15.8. Let $P \subset R'$ be the strong splitting of $R'$ over $u'$. By More on Groupoids, Lemma 40.9.1 the morphisms $s', t' : R' \to U'$ are flat and locally of finite presentation. They are finite by assumption. Hence $s', t'$ are finite locally free, see Morphisms, Lemma 29.48.2. In particular $t(s^{-1}(u'))$ is a finite set of points $\{ u'_1, u'_2, \ldots , u'_ n\} $ of $U'$. Choose a quasi-compact open $W \subset U'$ containing each $u'_ i$. As $U$ is affine the morphism $W \to U$ is quasi-compact (see Schemes, Lemma 26.19.2). The morphism $W \to U$ is also locally quasi-finite (see Morphisms, Lemma 29.36.6) and separated. Hence by More on Morphisms, Lemma 37.43.2 (a version of Zariski's Main Theorem) we conclude that $W$ is quasi-affine. By Properties, Lemma 28.29.5 we see that $\{ u'_1, \ldots , u'_ n\} $ are contained in an affine open of $U'$. Thus we may apply Groupoids, Lemma 39.24.1 to conclude that there exists an affine $P$-invariant open $U'' \subset U'$ which contains $u'$.
To finish the proof denote $R'' = R|_{U''}$ the restriction of $R$ to $U''$. This is the same as the restriction of $R'$ to $U''$. As $P \subset R'$ is an open and closed subscheme, so is $P|_{U''} \subset R''$. By construction the open subscheme $U'' \subset U'$ is $P$-invariant which means that $P|_{U''} = (s'|_ P)^{-1}(U'') = (t'|_ P)^{-1}(U'')$ (see discussion in Groupoids, Section 39.19) so the restrictions of $s''$ and $t''$ to $P|_{U''}$ are still finite. The sub groupoid scheme $P|_{U''}$ is still a strong splitting of $R''$ over $u''$; above we verified (a), (b) and (c) holds as $\{ r' \in R': t'(r') = u', s'(r') = u'\} = \{ r'' \in R'': t''(r'') = u', s''(r'') = u'\} $ trivially. The lemma is proved.
$\square$
Lemma 79.15.12. In Situation 79.15.3 assume in addition that $s, t$ are flat and locally of finite presentation and that $U$ is affine. Then there exists an affine scheme $U'$, an étale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa (u) = \kappa (u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is split over $u'$.
Proof.
The proof of this lemma is literally the same as the proof of Lemma 79.15.11 except that “strong splitting” needs to be replaced by “splitting” (2 times) and that the reference to Lemma 79.15.8 needs to be replaced by a reference to Lemma 79.15.9.
$\square$
Lemma 79.15.13. In Situation 79.15.4 assume in addition that $s, t$ are flat and locally of finite presentation and that $U$ is affine. Then there exists an affine scheme $U'$, an étale morphism $U' \to U$, and a point $u' \in U'$ lying over $u$ with $\kappa (u) = \kappa (u')$ such that the restriction $R' = R|_{U'}$ of $R$ to $U'$ is quasi-split over $u'$.
Proof.
The proof of this lemma is literally the same as the proof of Lemma 79.15.11 except that “strong splitting” needs to be replaced by “quasi-splitting” (2 times) and that the reference to Lemma 79.15.8 needs to be replaced by a reference to Lemma 79.15.10.
$\square$
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