92.15 Algebraic stacks, overhauled
Some basic results on algebraic stacks.
Lemma 92.15.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic stacks over $S$. Let $V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$. Let $y : (\mathit{Sch}/V)_{fppf} \to \mathcal{Y}$ be surjective and smooth. Then there exists an object $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a $2$-commutative diagram
\[ \xymatrix{ (\mathit{Sch}/U)_{fppf} \ar[r]_ a \ar[d]_ x & (\mathit{Sch}/V)_{fppf} \ar[d]^ y \\ \mathcal{X} \ar[r]^ f & \mathcal{Y} } \]
with $x$ surjective and smooth.
Proof.
First choose $W \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a surjective smooth $1$-morphism $z : (\mathit{Sch}/W)_{fppf} \to \mathcal{X}$. As $\mathcal{Y}$ is an algebraic stack we may choose an equivalence
\[ j : \mathcal{S}_ F \longrightarrow (\mathit{Sch}/W)_{fppf} \times _{f \circ z, \mathcal{Y}, y} (\mathit{Sch}/V)_{fppf} \]
where $F$ is an algebraic space. By Lemma 92.10.6 the morphism $\mathcal{S}_ F \to (\mathit{Sch}/W)_{fppf}$ is surjective and smooth as a base change of $y$. Hence by Lemma 92.10.5 we see that $\mathcal{S}_ F \to \mathcal{X}$ is surjective and smooth. Choose an object $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a surjective étale morphism $U \to F$. Then applying Lemma 92.10.5 once more we obtain the desired properties.
$\square$
This lemma is a generalization of Proposition 92.13.3.
Lemma 92.15.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic stacks over $S$. The following are equivalent:
for $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ the functor $f : \mathcal{X}_ U \to \mathcal{Y}_ U$ is faithful,
the functor $f$ is faithful, and
$f$ is representable by algebraic spaces.
Proof.
Parts (1) and (2) are equivalent by general properties of $1$-morphisms of categories fibred in groupoids, see Categories, Lemma 4.35.8. We see that (3) implies (2) by Lemma 92.9.2. Finally, assume (2). Let $U$ be a scheme. Let $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ U)$. We have to prove that
\[ \mathcal{W} = (\mathit{Sch}/U)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X} \]
is representable by an algebraic space over $U$. Since $(\mathit{Sch}/U)_{fppf}$ is an algebraic stack we see from Lemma 92.14.3 that $\mathcal{W}$ is an algebraic stack. On the other hand the explicit description of objects of $\mathcal{W}$ as triples $(V, x, \alpha : y(V) \to f(x))$ and the fact that $f$ is faithful, shows that the fibre categories of $\mathcal{W}$ are setoids. Hence Proposition 92.13.3 guarantees that $\mathcal{W}$ is representable by an algebraic space.
$\square$
Lemma 92.15.3. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $u : \mathcal{U} \to \mathcal{X}$ be a $1$-morphism of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. If
$\mathcal{U}$ is representable by an algebraic space, and
$u$ is representable by algebraic spaces, surjective and smooth,
then $\mathcal X$ is an algebraic stack over $S$.
Proof.
We have to show that $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces, see Definition 92.12.1. Given two schemes $T_1$, $T_2$ over $S$ denote $\mathcal{T}_ i = (\mathit{Sch}/T_ i)_{fppf}$ the associated representable fibre categories. Suppose given $1$-morphisms $f_ i : \mathcal{T}_ i \to \mathcal{X}$. According to Lemma 92.10.11 it suffices to prove that the $2$-fibered product $\mathcal{T}_1 \times _\mathcal {X} \mathcal{T}_2$ is representable by an algebraic space. By Stacks, Lemma 8.6.8 this is in any case a stack in setoids. Thus $\mathcal{T}_1 \times _\mathcal {X} \mathcal{T}_2$ corresponds to some sheaf $F$ on $(\mathit{Sch}/S)_{fppf}$, see Stacks, Lemma 8.6.3. Let $U$ be the algebraic space which represents $\mathcal{U}$. By assumption
\[ \mathcal{T}_ i' = \mathcal{U} \times _{u, \mathcal{X}, f_ i} \mathcal{T}_ i \]
is representable by an algebraic space $T'_ i$ over $S$. Hence $\mathcal{T}_1' \times _\mathcal {U} \mathcal{T}_2'$ is representable by the algebraic space $T'_1 \times _ U T'_2$. Consider the commutative diagram
\[ \xymatrix{ & \mathcal{T}_1 \times _{\mathcal X} \mathcal{T}_2 \ar[rr]\ar '[d][dd] & & \mathcal{T}_1 \ar[dd] \\ \mathcal{T}_1' \times _\mathcal {U} \mathcal{T}_2' \ar[ur]\ar[rr]\ar[dd] & & \mathcal{T}_1' \ar[ur]\ar[dd] \\ & \mathcal{T}_2 \ar '[r][rr] & & \mathcal X \\ \mathcal{T}_2' \ar[rr]\ar[ur] & & \mathcal{U} \ar[ur] } \]
In this diagram the bottom square, the right square, the back square, and the front square are $2$-fibre products. A formal argument then shows that $\mathcal{T}_1' \times _\mathcal {U} \mathcal{T}_2' \to \mathcal{T}_1 \times _{\mathcal X} \mathcal{T}_2$ is the “base change” of $\mathcal{U} \to \mathcal{X}$, more precisely the diagram
\[ \xymatrix{ \mathcal{T}_1' \times _\mathcal {U} \mathcal{T}_2' \ar[d] \ar[r] & \mathcal{U} \ar[d] \\ \mathcal{T}_1 \times _{\mathcal X} \mathcal{T}_2 \ar[r] & \mathcal{X} } \]
is a $2$-fibre square. Hence $T'_1 \times _ U T'_2 \to F$ is representable by algebraic spaces, smooth, and surjective, see Lemmas 92.9.6, 92.9.7, 92.10.4, and 92.10.6. Therefore $F$ is an algebraic space by Bootstrap, Theorem 78.10.1 and we win.
$\square$
An application of Lemma 92.15.3 is that something which is an algebraic space over an algebraic stack is an algebraic stack. This is the analogue of Bootstrap, Lemma 78.3.6. Actually, it suffices to assume the morphism $\mathcal{X} \to \mathcal{Y}$ is “algebraic”, as we will see in Criteria for Representability, Lemma 95.8.2.
Lemma 92.15.4. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X} \to \mathcal{Y}$ be a morphism of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume that
$\mathcal{X} \to \mathcal{Y}$ is representable by algebraic spaces, and
$\mathcal{Y}$ is an algebraic stack over $S$.
Then $\mathcal{X}$ is an algebraic stack over $S$.
Proof.
Let $\mathcal{V} \to \mathcal{Y}$ be a surjective smooth $1$-morphism from a representable stack in groupoids to $\mathcal{Y}$. This exists by Definition 92.12.1. Then the $2$-fibre product $\mathcal{U} = \mathcal{V} \times _{\mathcal Y} \mathcal X$ is representable by an algebraic space by Lemma 92.9.8. The $1$-morphism $\mathcal{U} \to \mathcal X$ is representable by algebraic spaces, smooth, and surjective, see Lemmas 92.9.7 and 92.10.6. By Lemma 92.15.3 we conclude that $\mathcal{X}$ is an algebraic stack.
$\square$
reference
Lemma 92.15.5. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $j : \mathcal X \to \mathcal Y$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume $j$ is representable by algebraic spaces. Then, if $\mathcal{Y}$ is a stack in groupoids (resp. an algebraic stack), so is $\mathcal{X}$.
Proof.
The statement on algebraic stacks will follow from the statement on stacks in groupoids by Lemma 92.15.4. If $j$ is representable by algebraic spaces, then $j$ is faithful on fibre categories and for each $U$ and each $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ U)$ the presheaf
\[ (h : V \to U) \longmapsto \{ (x, \phi ) \mid x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ V), \phi : h^*y \to f(x)\} /\cong \]
is an algebraic space over $U$. See Lemma 92.9.2. In particular this presheaf is a sheaf and the conclusion follows from Stacks, Lemma 8.6.11.
$\square$
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