Lemma 80.8.1. Let S be a scheme. Let j : R \to U \times _ S U be an equivalence relation on schemes over S. Assume s, t : R \to U are flat and locally of finite presentation. Then there exists an equivalence relation j' : R' \to U'\times _ S U' on schemes over S, and an isomorphism
U'/R' \longrightarrow U/R
induced by a morphism U' \to U which maps R' into R such that s', t' : R \to U are flat, locally of finite presentation and locally quasi-finite.
Proof.
We will prove this lemma in several steps. We will use without further mention that an equivalence relation gives rise to a groupoid scheme and that the restriction of an equivalence relation is an equivalence relation, see Groupoids, Lemmas 39.3.2, 39.13.3, and 39.18.3.
Step 1: We may assume that s, t : R \to U are locally of finite presentation and Cohen-Macaulay morphisms. Namely, as in More on Groupoids, Lemma 40.8.1 let g : U' \to U be the open subscheme such that t^{-1}(U') \subset R is the maximal open over which s : R \to U is Cohen-Macaulay, and denote R' the restriction of R to U'. By the lemma cited above we see that
\xymatrix{ t^{-1}(U') \ar@{=}[r] & U' \times _{g, U, t} R \ar[r]_-{\text{pr}_1} \ar@/^3ex/[rr]^ h & R \ar[r]_ s & U }
is surjective. Since h is flat and locally of finite presentation, we see that \{ h\} is a fppf covering. Hence by Groupoids, Lemma 39.20.6 we see that U'/R' \to U/R is an isomorphism. By the construction of U' we see that s', t' are Cohen-Macaulay and locally of finite presentation.
Step 2. Assume s, t are Cohen-Macaulay and locally of finite presentation. Let u \in U be a point of finite type. By More on Groupoids, Lemma 40.12.4 there exists an affine scheme U' and a morphism g : U' \to U such that
g is an immersion,
u \in U',
g is locally of finite presentation,
h is flat, locally of finite presentation and locally quasi-finite, and
the morphisms s', t' : R' \to U' are flat, locally of finite presentation and locally quasi-finite.
Here we have used the notation introduced in More on Groupoids, Situation 40.12.1.
Step 3. For each point u \in U which is of finite type choose a g_ u : U'_ u \to U as in Step 2 and denote R'_ u the restriction of R to U'_ u. Denote h_ u = s \circ \text{pr}_1 : U'_ u \times _{g_ u, U, t} R \to U. Set U' = \coprod _{u \in U} U'_ u, and g = \coprod g_ u. Let R' be the restriction of R to U' as above. We claim that the pair (U', g) works1. Note that
\begin{align*} R' = & \coprod \nolimits _{u_1, u_2 \in U} (U'_{u_1} \times _{g_{u_1}, U, t} R) \times _ R (R \times _{s, U, g_{u_2}} U'_{u_2}) \\ = & \coprod \nolimits _{u_1, u_2 \in U} (U'_{u_1} \times _{g_{u_1}, U, t} R) \times _{h_{u_1}, U, g_{u_2}} U'_{u_2} \end{align*}
Hence the projection s' : R' \to U' = \coprod U'_{u_2} is flat, locally of finite presentation and locally quasi-finite as a base change of \coprod h_{u_1}. Finally, by construction the morphism h : U' \times _{g, U, t} R \to U is equal to \coprod h_ u hence its image contains all points of finite type of U. Since each h_ u is flat and locally of finite presentation we conclude that h is flat and locally of finite presentation. In particular, the image of h is open (see Morphisms, Lemma 29.25.10) and since the set of points of finite type is dense (see Morphisms, Lemma 29.16.7) we conclude that the image of h is U. This implies that \{ h\} is an fppf covering. By Groupoids, Lemma 39.20.6 this means that U'/R' \to U/R is an isomorphism. This finishes the proof of the lemma.
\square
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