Proposition 64.14.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume

$s, t : R \to U$ finite locally free,

$j = (t, s)$ is an equivalence, and

for a dense set of points $u \in U$ the $R$-equivalence class $t(s^{-1}(\{ u\} ))$ is contained in an affine open of $U$.

Then there exists a finite locally free morphism $U \to M$ of schemes over $S$ such that $R = U \times _ M U$ and such that $M$ represents the quotient sheaf $U/R$ in the fppf topology.

**Proof.**
By assumption (3) and Groupoids, Lemma 39.24.1 we can find an open covering $U = \bigcup U_ i$ such that each $U_ i$ is an $R$-invariant affine open of $U$. Set $R_ i = R|_{U_ i}$. Consider the fppf sheaves $F = U/R$ and $F_ i = U_ i/R_ i$. By Spaces, Lemma 63.10.2 the morphisms $F_ i \to F$ are representable and open immersions. By Groupoids, Proposition 39.23.9 the sheaves $F_ i$ are representable by affine schemes. If $T$ is a scheme and $T \to F$ is a morphism, then $V_ i = F_ i \times _ F T$ is open in $T$ and we claim that $T = \bigcup V_ i$. Namely, fppf locally on $T$ we can lift $T \to F$ to a morphism $f : T \to U$ and in that case $f^{-1}(U_ i) \subset V_ i$. Hence we conclude that $F$ is representable by a scheme, see Schemes, Lemma 26.15.4.
$\square$

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