Lemma 97.17.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Then the morphism $\mathcal{S}_ U \to [U/R]$ is flat, locally of finite presentation, and surjective.
97.17 Applications
Our first task is to show that the quotient stack $[U/R]$ associated to a “flat and locally finitely presented groupoid” is an algebraic stack. See Groupoids in Spaces, Definition 78.20.1 for the definition of the quotient stack. The following lemma is preliminary and is the analogue of Algebraic Stacks, Lemma 94.17.2.
Proof. Let $T$ be a scheme and let $x : (\mathit{Sch}/T)_{fppf} \to [U/R]$ be a $1$-morphism. We have to show that the projection
\[ \mathcal{S}_ U \times _{[U/R]} (\mathit{Sch}/T)_{fppf} \longrightarrow (\mathit{Sch}/T)_{fppf} \]
is surjective, flat, and locally of finite presentation. We already know that the left hand side is representable by an algebraic space $F$, see Algebraic Stacks, Lemmas 94.17.1 and 94.10.11. Hence we have to show the corresponding morphism $F \to T$ of algebraic spaces is surjective, locally of finite presentation, and flat. Since we are working with properties of morphisms of algebraic spaces which are local on the target in the fppf topology we may check this fppf locally on $T$. By construction, there exists an fppf covering $\{ T_ i \to T\} $ of $T$ such that $x|_{(\mathit{Sch}/T_ i)_{fppf}}$ comes from a morphism $x_ i : T_ i \to U$. (Note that $F \times _ T T_ i$ represents the $2$-fibre product $\mathcal{S}_ U \times _{[U/R]} (\mathit{Sch}/T_ i)_{fppf}$ so everything is compatible with the base change via $T_ i \to T$.) Hence we may assume that $x$ comes from $x : T \to U$. In this case we see that
\[ \mathcal{S}_ U \times _{[U/R]} (\mathit{Sch}/T)_{fppf} = (\mathcal{S}_ U \times _{[U/R]} \mathcal{S}_ U) \times _{\mathcal{S}_ U} (\mathit{Sch}/T)_{fppf} = \mathcal{S}_ R \times _{\mathcal{S}_ U} (\mathit{Sch}/T)_{fppf} \]
The first equality by Categories, Lemma 4.31.10 and the second equality by Groupoids in Spaces, Lemma 78.22.2. Clearly the last $2$-fibre product is represented by the algebraic space $F = R \times _{s, U, x} T$ and the projection $R \times _{s, U, x} T \to T$ is flat and locally of finite presentation as the base change of the flat locally finitely presented morphism of algebraic spaces $s : R \to U$. It is also surjective as $s$ has a section (namely the identity $e : U \to R$ of the groupoid). This proves the lemma. $\square$
Here is the first main result of this section.
Theorem 97.17.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Then the quotient stack $[U/R]$ is an algebraic stack over $S$.
Proof. We check the two conditions of Theorem 97.16.1 for the morphism
\[ (\mathit{Sch}/U)_{fppf} \longrightarrow [U/R]. \]
The first is trivial (as $U$ is an algebraic space). The second is Lemma 97.17.1. $\square$
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