97.18 When is a quotient stack algebraic?
In Groupoids in Spaces, Section 78.20 we have defined the quotient stack $[U/R]$ associated to a groupoid $(U, R, s, t, c)$ in algebraic spaces. Note that $[U/R]$ is a stack in groupoids whose diagonal is representable by algebraic spaces (see Bootstrap, Lemma 80.11.5 and Algebraic Stacks, Lemma 94.10.11) and such that there exists an algebraic space $U$ and a $1$-morphism $(\mathit{Sch}/U)_{fppf} \to [U/R]$ which is an “fppf surjection” in the sense that it induces a map on presheaves of isomorphism classes of objects which becomes surjective after sheafification. However, it is not the case that $[U/R]$ is an algebraic stack in general. This is not a contradiction with Theorem 97.16.1 as the $1$-morphism $(\mathit{Sch}/U)_{fppf} \to [U/R]$ may not be flat and locally of finite presentation.
The easiest way to make examples of non-algebraic quotient stacks is to look at quotients of the form $[S/G]$ where $S$ is a scheme and $G$ is a group scheme over $S$ acting trivially on $S$. Namely, we will see below (Lemma 97.18.3) that if $[S/G]$ is algebraic, then $G \to S$ has to be flat and locally of finite presentation. An explicit example can be found in Examples, Section 110.52.
Lemma 97.18.1. Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The quotient stack $[U/R]$ is an algebraic stack if and only if there exists a morphism of algebraic spaces $g : U' \to U$ such that
the composition $U' \times _{g, U, t} R \to R \xrightarrow {s} U$ is a surjection of sheaves, and
the morphisms $s', t' : R' \to U'$ are flat and locally of finite presentation where $(U', R', s', t', c')$ is the restriction of $(U, R, s, t, c)$ via $g$.
Proof.
First, assume that $g : U' \to U$ satisfies (1) and (2). Property (1) implies that $[U'/R'] \to [U/R]$ is an equivalence, see Groupoids in Spaces, Lemma 78.25.2. By Theorem 97.17.2 the quotient stack $[U'/R']$ is an algebraic stack. Hence $[U/R]$ is an algebraic stack too, see Algebraic Stacks, Lemma 94.12.4.
Conversely, assume that $[U/R]$ is an algebraic stack. We may choose a scheme $W$ and a surjective smooth $1$-morphism
\[ f : (\mathit{Sch}/W)_{fppf} \longrightarrow [U/R]. \]
By the $2$-Yoneda lemma (Algebraic Stacks, Section 94.5) this corresponds to an object $\xi $ of $[U/R]$ over $W$. By the description of $[U/R]$ in Groupoids in Spaces, Lemma 78.24.1 we can find a surjective, flat, locally finitely presented morphism $b : U' \to W$ of schemes such that $\xi ' = b^*\xi $ corresponds to a morphism $g : U' \to U$. Note that the $1$-morphism
\[ f' : (\mathit{Sch}/U')_{fppf} \longrightarrow [U/R]. \]
corresponding to $\xi '$ is surjective, flat, and locally of finite presentation, see Algebraic Stacks, Lemma 94.10.5. Hence $(\mathit{Sch}/U')_{fppf} \times _{[U/R]} (\mathit{Sch}/U')_{fppf}$ which is represented by the algebraic space
\[ \mathit{Isom}_{[U/R]}(\text{pr}_0^*\xi ', \text{pr}_1^*\xi ') = (U' \times _ S U') \times _{(g \circ \text{pr}_0, g \circ \text{pr}_1), U \times _ S U} R = R' \]
(see Groupoids in Spaces, Lemma 78.22.1 for the first equality; the second is the definition of restriction) is flat and locally of finite presentation over $U'$ via both $s'$ and $t'$ (by base change, see Algebraic Stacks, Lemma 94.10.6). By this description of $R'$ and by Algebraic Stacks, Lemma 94.16.1 we obtain a canonical fully faithful $1$-morphism $[U'/R'] \to [U/R]$. This $1$-morphism is essentially surjective because $f'$ is flat, locally of finite presentation, and surjective (see Stacks, Lemma 8.4.8); another way to prove this is to use Algebraic Stacks, Remark 94.16.3. Finally, we can use Groupoids in Spaces, Lemma 78.25.2 to conclude that the composition $U' \times _{g, U, t} R \to R \xrightarrow {s} U$ is a surjection of sheaves.
$\square$
Lemma 97.18.2. Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$ and let $a : G \times _ B X \to X$ be an action of $G$ on $X$ over $B$. The quotient stack $[X/G]$ is an algebraic stack if and only if there exists a morphism of algebraic spaces $\varphi : X' \to X$ such that
$G \times _ B X' \to X$, $(g, x') \mapsto a(g, \varphi (x'))$ is a surjection of sheaves, and
the two projections $X'' \to X'$ of the algebraic space $X''$ given by the rule
\[ T \longmapsto \{ (x'_1, g, x'_2) \in (X' \times _ B G \times _ B X')(T) \mid \varphi (x'_1) = a(g, \varphi (x'_2))\} \]
are flat and locally of finite presentation.
Proof.
This lemma is a special case of Lemma 97.18.1. Namely, the quotient stack $[X/G]$ is by Groupoids in Spaces, Definition 78.20.1 equal to the quotient stack $[X/G \times _ B X]$ of the groupoid in algebraic spaces $(X, G \times _ B X, s, t, c)$ associated to the group action in Groupoids in Spaces, Lemma 78.15.1. There is one small observation that is needed to get condition (1). Namely, the morphism $s : G \times _ B X \to X$ is the second projection and the morphism $t : G \times _ B X \to X$ is the action morphism $a$. Hence the morphism $h : U' \times _{g, U, t} R \to R \xrightarrow {s} U$ from Lemma 97.18.1 corresponds to the morphism
\[ X' \times _{\varphi , X, a} (G \times _ B X) \xrightarrow {\text{pr}_1} X \]
in the current setting. However, because of the symmetry given by the inverse of $G$ this morphism is isomorphic to the morphism
\[ (G \times _ B X) \times _{\text{pr}_1, X, \varphi } X' \xrightarrow {a} X \]
of the statement of the lemma. Details omitted.
$\square$
slogan
Lemma 97.18.3. Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Endow $B$ with the trivial action of $G$. Then the quotient stack $[B/G]$ is an algebraic stack if and only if $G$ is flat and locally of finite presentation over $B$.
Proof.
If $G$ is flat and locally of finite presentation over $B$, then $[B/G]$ is an algebraic stack by Theorem 97.17.2.
Conversely, assume that $[B/G]$ is an algebraic stack. By Lemma 97.18.2 and because the action is trivial, we see there exists an algebraic space $B'$ and a morphism $B' \to B$ such that (1) $B' \to B$ is a surjection of sheaves and (2) the projections
\[ B' \times _ B G \times _ B B' \to B' \]
are flat and locally of finite presentation. Note that the base change $B' \times _ B G \times _ B B' \to G \times _ B B'$ of $B' \to B$ is a surjection of sheaves also. Thus it follows from Descent on Spaces, Lemma 74.8.1 that the projection $G \times _ B B' \to B'$ is flat and locally of finite presentation. By (1) we can find an fppf covering $\{ B_ i \to B\} $ such that $B_ i \to B$ factors through $B' \to B$. Hence $G \times _ B B_ i \to B_ i$ is flat and locally of finite presentation by base change. By Descent on Spaces, Lemmas 74.11.13 and 74.11.10 we conclude that $G \to B$ is flat and locally of finite presentation.
$\square$
Later we will see that the quotient stack of a smooth $S$-space by a group algebraic space $G$ is smooth, even when $G$ is not smooth (Morphisms of Stacks, Lemma 101.33.7).
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