The Stacks project

Proof. It is clear from the description of $s', t'$ in Groupoids, Lemma 39.18.1 that $s' , t' : R' \to U'$ are étale as compositions of base changes of étale morphisms (see Morphisms, Lemma 29.36.4 and 29.36.3). $\square$

Proof. By Groupoids, Lemma 39.3.2 the morphism $j' = (s', t') : R' \to U' \times _ S U'$ defines an equivalence relation. Since $g$ is flat and locally of finite presentation we see that $g$ is universally open as well (Morphisms, Lemma 29.25.10). For the same reason $s, t$ are universally open as well. Let $W^1 = g(U') \subset U$, and let $W = t(s^{-1}(W^1))$. Then $W^1$ and $W$ are open in $U$. Moreover, as $j$ is an equivalence relation we have $t(s^{-1}(W)) = W$ (see Groupoids, Lemma 39.19.2 for example).

By Groupoids, Lemma 39.20.5 the map of sheaves $F' = U'/R' \to F = U/R$ is injective. Let $a : T \to F$ be a morphism from a scheme into $U/R$. We have to show that $T \times _ F F'$ is representable by an open subscheme of $T$.

The morphism $a$ is given by the following data: an fppf covering $\{ \varphi _ j : T_ j \to T\} _{j \in J}$ of $T$ and morphisms $a_ j : T_ j \to U$ such that the maps

factor through $j : R \to U \times _ S U$ via some (unique) maps $r_{jj'} : T_ j \times _ T T_{j'} \to R$. The system $(a_ j)$ corresponds to $a$ in the sense that the diagrams

commute.

Consider the open subsets $W_ j = a_ j^{-1}(W) \subset T_ j$. Since $t(s^{-1}(W)) = W$ we see that

By Descent, Lemma 35.13.6 this means there exists an open $W_ T \subset T$ such that $\varphi _ j^{-1}(W_ T) = W_ j$ for all $j \in J$. We claim that $W_ T \to T$ represents $T \times _ F F' \to T$.

First, let us show that $W_ T \to T \to F$ is an element of $F'(W_ T)$. Since $\{ W_ j \to W_ T\} _{j \in J}$ is an fppf covering of $W_ T$, it is enough to show that each $W_ j \to U \to F$ is an element of $F'(W_ j)$ (as $F'$ is a sheaf for the fppf topology). Consider the commutative diagram

where $W'_ j = W_ j \times _ W s^{-1}(W^1) \times _{W^1} U'$. Since $t$ and $g$ are surjective, flat and locally of finite presentation, so is $W'_ j \to W_ j$. Hence the restriction of the element $W_ j \to U \to F$ to $W'_ j$ is an element of $F'$ as desired.

Suppose that $f : T' \to T$ is a morphism of schemes such that $a|_{T'} \in F'(T')$. We have to show that $f$ factors through the open $W_ T$. Since $\{ T' \times _ T T_ j \to T'\} $ is an fppf covering of $T'$ it is enough to show each $T' \times _ T T_ j \to T$ factors through $W_ T$. Hence we may assume $f$ factors as $\varphi _ j \circ f_ j : T' \to T_ j \to T$ for some $j$. In this case the condition $a|_{T'} \in F'(T')$ means that there exists some fppf covering $\{ \psi _ i : T'_ i \to T'\} _{i \in I}$ and some morphisms $b_ i : T'_ i \to U'$ such that

is commutative. This commutativity means that there exists a morphism $r'_ i : T'_ i \to R$ such that $t \circ r'_ i = a_ j \circ f_ j \circ \psi _ i$, and $s \circ r'_ i = g \circ b_ i$. This implies that $\mathop{\mathrm{Im}}(f_ j \circ \psi _ i) \subset W_ j$ and we win. $\square$

Proof. Denote $c : U \to U/R$ the morphism in question. Let $T$ be a scheme and let $a : T \to U/R$ be a morphism. We have to show that the morphism (of schemes) $\pi : T \times _{a, U/R, c} U \to T$ is étale and surjective. The morphism $a$ corresponds to an fppf covering $\{ \varphi _ i : T_ i \to T\} $ and morphisms $a_ i : T_ i \to U$ such that $a_ i \times a_{i'} : T_ i \times _ T T_{i'} \to U \times _ S U$ factors through $R$, and such that $c \circ a_ i = a \circ \varphi _ i$. Hence

Since $t$ is étale and surjective we conclude that the base change of $\pi $ to $T_ i$ is surjective and étale. Since the property of being surjective and étale is local on the base in the fpqc topology (see Remark 65.4.3) we win. $\square$

Proof. Since $j : R \to U \times _ S U$ is a monomorphism we see that $j$ is separated (see Schemes, Lemma 26.23.3). Since $U$ is affine we see that $U \times _ S U$ (which comes equipped with a monomorphism into the affine scheme $U \times U$) is separated. Hence we see that $R$ is separated. In particular the morphisms $s, t$ are separated as well as étale.

Since the composition $R \to U \times _ S U \to U$ is locally of finite type we conclude that $j$ is locally of finite type (see Morphisms, Lemma 29.15.8). As $j$ is also a monomorphism it has finite fibres and we see that $j$ is locally quasi-finite by Morphisms, Lemma 29.20.7. Altogether we see that $j$ is separated and locally quasi-finite.

Our first step is to show that the quotient map $c : U \to F$ is representable. Consider a scheme $T$ and a morphism $a : T \to F$. We have to show that the sheaf $G = T \times _{a, F, c} U$ is representable. As seen in the proofs of Lemmas 65.10.2 and 65.10.3 there exists an fppf covering $\{ \varphi _ i : T_ i \to T\} _{i \in I}$ and morphisms $a_ i : T_ i \to U$ such that $a_ i \times a_{i'} : T_ i \times _ T T_{i'} \to U \times _ S U$ factors through $R$, and such that $c \circ a_ i = a \circ \varphi _ i$. As in the proof of Lemma 65.10.3 we see that

Since $t$ is separated and étale, and in particular separated and locally quasi-finite (by Morphisms, Lemmas 29.35.10 and 29.36.16) we see that the restriction of $G$ to each $T_ i$ is representable by a morphism of schemes $X_ i \to T_ i$ which is separated and locally quasi-finite. By Descent, Lemma 35.39.1 we obtain a descent datum $(X_ i, \varphi _{ii'})$ relative to the fppf-covering $\{ T_ i \to T\} $. Since each $X_ i \to T_ i$ is separated and locally quasi-finite we see by More on Morphisms, Lemma 37.57.1 that this descent datum is effective. Hence by Descent, Lemma 35.39.1 (2) we conclude that $G$ is representable as desired.

The second step of the proof is to show that $U \to F$ is surjective and étale. This is clear from the above since in the first step above we saw that $G = T \times _{a, F, c} U$ is a scheme over $T$ which base changes to schemes $X_ i \to T_ i$ which are surjective and étale. Thus $G \to T$ is surjective and étale (see Remark 65.4.3). Alternatively one can reread the proof of Lemma 65.10.3 in the current situation.

The third and final step is to show that the diagonal map $F \to F \times F$ is representable. We first observe that the diagram

is a fibre product square. By Lemma 65.3.4 the morphism $U \times _ S U \to F \times F$ is representable (note that $h_ U \times h_ U = h_{U \times _ S U}$). Moreover, by Lemma 65.5.7 the morphism $U \times _ S U \to F \times F$ is surjective and étale (note also that étale and surjective occur in the lists of Remarks 65.4.3 and 65.4.2). It follows either from Lemma 65.3.3 and the diagram above, or by writing $R \to F$ as $R \to U \to F$ and Lemmas 65.3.1 and 65.3.2 that $R \to F$ is representable as well. Let $T$ be a scheme and let $a : T \to F \times F$ be a morphism. We have to show that $G = T \times _{a, F \times F, \Delta } F$ is representable. By what was said above the morphism (of schemes)

is surjective and étale. Hence $\{ T' \to T\} $ is an étale covering of $T$. Note also that

as can be seen contemplating the following cube

Hence we see that the restriction of $G$ to $T'$ is representable by a scheme $X$, and moreover that the morphism $X \to T'$ is a base change of the morphism $j$. Hence $X \to T'$ is separated and locally quasi-finite (see second paragraph of the proof). By Descent, Lemma 35.39.1 we obtain a descent datum $(X, \varphi )$ relative to the fppf-covering $\{ T' \to T\} $. Since $X \to T'$ is separated and locally quasi-finite we see by More on Morphisms, Lemma 37.57.1 that this descent datum is effective. Hence by Descent, Lemma 35.39.1 (2) we conclude that $G$ is representable as desired. $\square$

Proof. By Lemma 65.10.3 it suffices to prove that $U/R$ is an algebraic space. Let $U' \to U$ be a surjective, étale morphism. Then $\{ U' \to U\} $ is in particular an fppf covering. Let $R'$ be the restriction of $R$ to $U'$, see Groupoids, Definition 39.3.3. According to Groupoids, Lemma 39.20.6 we see that $U/R \cong U'/R'$. By Lemma 65.10.1 $R'$ is an étale equivalence relation on $U'$. Thus we may replace $U$ by $U'$.

We apply the previous remark to $U' = \coprod U_ i$, where $U = \bigcup U_ i$ is an affine open covering of $U$. Hence we may and do assume that $U = \coprod U_ i$ where each $U_ i$ is an affine scheme.

Consider the restriction $R_ i$ of $R$ to $U_ i$. By Lemma 65.10.1 this is an étale equivalence relation. Set $F_ i = U_ i/R_ i$ and $F = U/R$. It is clear that $\coprod F_ i \to F$ is surjective. By Lemma 65.10.2 each $F_ i \to F$ is representable, and an open immersion. By Lemma 65.10.4 applied to $(U_ i, R_ i)$ we see that $F_ i$ is an algebraic space. Then by Lemma 65.10.3 we see that $U_ i \to F_ i$ is étale and surjective. From Lemma 65.8.4 it follows that $\coprod F_ i$ is an algebraic space. Finally, we have verified all hypotheses of Lemma 65.8.5 and it follows that $F = U/R$ is an algebraic space. $\square$