The Stacks project

93.17 The algebraic stack associated to a smooth groupoid

In this section we start with a smooth groupoid in algebraic spaces and we show that the associated quotient stack is an algebraic stack.

Lemma 93.17.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Then the diagonal of $[U/R]$ is representable by algebraic spaces.

Proof. It suffices to show that the $\mathit{Isom}$-sheaves are algebraic spaces, see Lemma 93.10.11. This follows from Bootstrap, Lemma 79.11.5. $\square$

Lemma 93.17.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces over $S$. Then the morphism $\mathcal{S}_ U \to [U/R]$ is smooth and surjective.

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 and smooth. We already know that the left hand side is representable by an algebraic space $F$, see Lemmas 93.17.1 and 93.10.11. Hence we have to show the corresponding morphism $F \to T$ of algebraic spaces is surjective and smooth. 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 77.22.2. 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 smooth as the base change of the smooth 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 main result of this section.

Theorem 93.17.3. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces over $S$. Then the quotient stack $[U/R]$ is an algebraic stack over $S$.

Proof. We check the three conditions of Definition 93.12.1. By construction we have that $[U/R]$ is a stack in groupoids which is the first condition.

The second condition follows from the stronger Lemma 93.17.1.

Finally, we have to show there exists a scheme $W$ over $S$ and a surjective smooth $1$-morphism $(\mathit{Sch}/W)_{fppf} \longrightarrow \mathcal{X}$. First choose $W \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a surjective ├ętale morphism $W \to U$. Note that this gives a surjective ├ętale morphism $\mathcal{S}_ W \to \mathcal{S}_ U$ of categories fibred in sets, see Lemma 93.10.3. Of course then $\mathcal{S}_ W \to \mathcal{S}_ U$ is also surjective and smooth, see Lemma 93.10.9. Hence $\mathcal{S}_ W \to \mathcal{S}_ U \to [U/R]$ is surjective and smooth by a combination of Lemmas 93.17.2 and 93.10.5. $\square$

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