The Stacks project

Lemma 63.10.4. Let $S$ be a scheme. Let $U$ be a scheme over $S$. Let $j = (s, t) : R \to U \times _ S U$ be an étale equivalence relation on $U$ over $S$. Assume that $U$ is affine. Then the quotient $F = U/R$ is an algebraic space, and $U \to F$ is étale and surjective.

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 63.10.2 and 63.10.3 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 63.10.3 we see that

\begin{eqnarray*} T_ i \times _{\varphi _ i, T} G & = & T_ i \times _{\varphi _ i, T} T \times _{a, U/R, c} U \\ & = & T_ i \times _{c \circ a_ i, U/R, c} U \\ & = & T_ i \times _{a_ i, U} U \times _{c, U/R, c} U \\ & = & T_ i \times _{a_ i, U, t} R \end{eqnarray*}

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 35.36.1 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 37.50.1 that this descent datum is effective. Hence by Descent, Lemma 35.36.1 (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 63.4.3). Alternatively one can reread the proof of Lemma 63.10.3 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

\[ \xymatrix{ R \ar[r] \ar[d]_ j & F \ar[d]^\Delta \\ U \times _ S U \ar[r] & F \times F } \]

is a fibre product square. By Lemma 63.3.4 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 63.5.7 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 63.4.3 and 63.4.2). It follows either from Lemma 63.3.3 and the diagram above, or by writing $R \to F$ as $R \to U \to F$ and Lemmas 63.3.1 and 63.3.2 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)

\[ T' = (U \times _ S U) \times _{F \times F, a} T \longrightarrow T \]

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

\[ T' \times _ T G = T' \times _{U \times _ S U, j} R \]

as can be seen contemplating the following cube

\[ \xymatrix{ & R \ar[rr] \ar[dd] & & F \ar[dd] \\ T' \times _ T G \ar[rr] \ar[dd] \ar[ru] & & G \ar[dd] \ar[ru] & \\ & U \times _ S U \ar '[r][rr] & & F \times F \\ T' \ar[rr] \ar[ru] & & T \ar[ru] } \]

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 35.36.1 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 37.50.1 that this descent datum is effective. Hence by Descent, Lemma 35.36.1 (2) we conclude that $G$ is representable as desired. $\square$


Comments (1)

Comment #5909 by Souparna Purohit on

Second to last line of the proof: "Since is separated..." should say "Since is..."

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