The Stacks project

90.13 Algebraic stacks and algebraic spaces

In this section we discuss some simple criteria which imply that an algebraic stack is an algebraic space. The main result is that this happens exactly when objects of fibre categories have no nontrivial automorphisms. This is not a triviality! Before we come to this we first do a sanity check.

Lemma 90.13.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$.

  1. A category fibred in groupoids $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ which is representable by an algebraic space is a Deligne-Mumford stack.

  2. If $F$ is an algebraic space over $S$, then the associated category fibred in groupoids $p : \mathcal{S}_ F \to (\mathit{Sch}/S)_{fppf}$ is a Deligne-Mumford stack.

  3. If $X \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, then $(\mathit{Sch}/X)_{fppf} \to (\mathit{Sch}/S)_{fppf}$ is a Deligne-Mumford stack.

Proof. It is clear that (2) implies (3). Parts (1) and (2) are equivalent by Lemma 90.12.4. Hence it suffices to prove (2). First, we note that $\mathcal{S}_ F$ is stack in sets since $F$ is a sheaf (Stacks, Lemma 8.6.3). A fortiori it is a stack in groupoids. Second the diagonal morphism $\mathcal{S}_ F \to \mathcal{S}_ F \times \mathcal{S}_ F$ is the same as the morphism $\mathcal{S}_ F \to \mathcal{S}_{F \times F}$ which comes from the diagonal of $F$. Hence this is representable by algebraic spaces according to Lemma 90.9.4. Actually it is even representable (by schemes), as the diagonal of an algebraic space is representable, but we do not need this. Let $U$ be a scheme and let $h_ U \to F$ be a surjective étale morphism. We may think of this a surjective étale morphism of algebraic spaces. Hence by Lemma 90.10.3 the corresponding $1$-morphism $(\mathit{Sch}/U)_{fppf} \to \mathcal{S}_ F$ is surjective and étale. $\square$

The following result says that a Deligne-Mumford stack whose inertia is trivial “is” an algebraic space. This lemma will be obsoleted by the stronger Proposition 90.13.3 below which says that this holds more generally for algebraic stacks...

Lemma 90.13.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}$ be an algebraic stack over $S$. The following are equivalent

  1. $\mathcal{X}$ is a Deligne-Mumford stack and is a stack in setoids,

  2. $\mathcal{X}$ is a Deligne-Mumford stack such that the canonical $1$-morphism $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is an equivalence, and

  3. $\mathcal{X}$ is representable by an algebraic space.

Proof. The equivalence of (1) and (2) follows from Stacks, Lemma 8.7.2. The implication (3) $\Rightarrow $ (1) follows from Lemma 90.13.1. Finally, assume (1). By Stacks, Lemma 8.6.3 there exists a sheaf $F$ on $(\mathit{Sch}/S)_{fppf}$ and an equivalence $j : \mathcal{X} \to \mathcal{S}_ F$. By Lemma 90.9.5 the fact that $\Delta _\mathcal {X}$ is representable by algebraic spaces, means that $\Delta _ F : F \to F \times F$ is representable by algebraic spaces. Let $U$ be a scheme, and let $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ be a surjective étale morphism. The composition $j \circ x : (\mathit{Sch}/U)_{fppf} \to \mathcal{S}_ F$ corresponds to a morphism $h_ U \to F$ of sheaves. By Bootstrap, Lemma 76.5.1 this morphism is representable by algebraic spaces. Hence by Lemma 90.10.4 we conclude that $h_ U \to F$ is surjective and étale. Finally, we apply Bootstrap, Theorem 76.6.1 to see that $F$ is an algebraic space. $\square$

Proposition 90.13.3. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}$ be an algebraic stack over $S$. The following are equivalent

  1. $\mathcal{X}$ is a stack in setoids,

  2. the canonical $1$-morphism $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is an equivalence, and

  3. $\mathcal{X}$ is representable by an algebraic space.

Proof. The equivalence of (1) and (2) follows from Stacks, Lemma 8.7.2. The implication (3) $\Rightarrow $ (1) follows from Lemma 90.13.2. Finally, assume (1). By Stacks, Lemma 8.6.3 there exists an equivalence $j : \mathcal{X} \to \mathcal{S}_ F$ where $F$ is a sheaf on $(\mathit{Sch}/S)_{fppf}$. By Lemma 90.9.5 the fact that $\Delta _\mathcal {X}$ is representable by algebraic spaces, means that $\Delta _ F : F \to F \times F$ is representable by algebraic spaces. Let $U$ be a scheme and let $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ be a surjective smooth morphism. The composition $j \circ x : (\mathit{Sch}/U)_{fppf} \to \mathcal{S}_ F$ corresponds to a morphism $h_ U \to F$ of sheaves. By Bootstrap, Lemma 76.5.1 this morphism is representable by algebraic spaces. Hence by Lemma 90.10.4 we conclude that $h_ U \to F$ is surjective and smooth. In particular it is surjective, flat and locally of finite presentation (by Lemma 90.10.9 and the fact that a smooth morphism of algebraic spaces is flat and locally of finite presentation, see Morphisms of Spaces, Lemmas 63.37.5 and 63.37.7). Finally, we apply Bootstrap, Theorem 76.10.1 to see that $F$ is an algebraic space. $\square$


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