The Stacks project

97.17 Algebraic stacks

Proposition 97.17.2 is our first main result on algebraic stacks.

Lemma 97.17.1. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Assume that

  1. $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces,

  2. $\mathcal{X}$ satisfies axioms [-1], [0], [1], [2], [3] (see Section 97.14),

  3. every formal object of $\mathcal{X}$ is effective,

  4. $\mathcal{X}$ satisfies openness of versality, and

  5. $\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$.

Then $\mathcal{X}$ is an algebraic stack.

Proof. Lemma 97.13.8 applies to $\mathcal{X}$. Using this we choose, for every finite type field $k$ over $S$ and every isomorphism class of object $x_0 \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_{\mathop{\mathrm{Spec}}(k)})$, an affine scheme $U_{k, x_0}$ of finite type over $S$ and a smooth morphism $(\mathit{Sch}/U_{k, x_0})_{fppf} \to \mathcal{X}$ such that there exists a finite type point $u_{k, x_0} \in U_{k, x_0}$ with residue field $k$ such that $x_0$ is the image of $u_{k, x_0}$. Then

\[ (\mathit{Sch}/U)_{fppf} \to \mathcal{X}, \quad \text{with}\quad U = \coprod \nolimits _{k, x_0} U_{k, x_0} \]

is smooth1. To finish the proof it suffices to show this map is surjective, see Criteria for Representability, Lemma 96.19.1 (this is where we use axiom [0]). By Criteria for Representability, Lemma 96.5.6 it suffices to show that $(\mathit{Sch}/U)_{fppf} \times _\mathcal {X} (\mathit{Sch}/V)_{fppf} \to (\mathit{Sch}/V)_{fppf}$ is surjective for those $y : (\mathit{Sch}/V)_{fppf} \to \mathcal{X}$ where $V$ is an affine scheme locally of finite presentation over $S$. By assumption (1) the fibre product $(\mathit{Sch}/U)_{fppf} \times _\mathcal {X} (\mathit{Sch}/V)_{fppf}$ is representable by an algebraic space $W$. Then $W \to V$ is smooth, hence the image is open. Hence it suffices to show that the image of $W \to V$ contains all finite type points of $V$, see Morphisms, Lemma 29.16.7. Let $v_0 \in V$ be a finite type point. Then $k = \kappa (v_0)$ is a finite type field over $S$. Denote $x_0 = y|_{\mathop{\mathrm{Spec}}(k)}$ the pullback of $y$ by $v_0$. Then $(u_{k, x_0}, v_0)$ will give a morphism $\mathop{\mathrm{Spec}}(k) \to W$ whose composition with $W \to V$ is $v_0$ and we win. $\square$

Proposition 97.17.2. Let $S$ be a locally Noetherian scheme. Let $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. Assume that

  1. $\Delta _\Delta : \mathcal{X} \to \mathcal{X} \times _{\mathcal{X} \times \mathcal{X}} \mathcal{X}$ is representable by algebraic spaces,

  2. $\mathcal{X}$ satisfies axioms [-1], [0], [1], [2], [3], [4], and [5] (see Section 97.14),

  3. $\mathcal{O}_{S, s}$ is a G-ring for all finite type points $s$ of $S$.

Then $\mathcal{X}$ is an algebraic stack.

Proof. We first prove that $\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ is representable by algebraic spaces. To do this it suffices to show that

\[ \mathcal{Y} = \mathcal{X} \times _{\Delta , \mathcal{X} \times \mathcal{X}, y} (\mathit{Sch}/V)_{fppf} \]

is representable by an algebraic space for any affine scheme $V$ locally of finite presentation over $S$ and object $y$ of $\mathcal{X} \times \mathcal{X}$ over $V$, see Criteria for Representability, Lemma 96.5.52. Observe that $\mathcal{Y}$ is fibred in setoids (Stacks, Lemma 8.2.5) and let $Y : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$, $T \mapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ T)/\cong $ be the functor of isomorphism classes. We will apply Proposition 97.16.1 to see that $Y$ is an algebraic space.

Note that $\Delta _\mathcal {Y} : \mathcal{Y} \to \mathcal{Y} \times \mathcal{Y}$ (and hence also $Y \to Y \times Y$) is representable by algebraic spaces by condition (1) and Criteria for Representability, Lemma 96.4.4. Observe that $Y$ is a sheaf for the ├ętale topology by Stacks, Lemmas 8.6.3 and 8.6.7, i.e., axiom [0] holds. Also $Y$ is limit preserving by Lemma 97.11.2, i.e., we have [1]. Note that $Y$ has (RS), i.e., axiom [2] holds, by Lemmas 97.5.2 and 97.5.3. Axiom [3] for $Y$ follows from Lemmas 97.8.1 and 97.8.2. Axiom [4] follows from Lemmas 97.9.5 and 97.9.6. Axiom [5] for $Y$ follows directly from openness of versality for $\Delta _\mathcal {X}$ which is part of axiom [5] for $\mathcal{X}$. Thus all the assumptions of Proposition 97.16.1 are satisfied and $Y$ is an algebraic space.

At this point it follows from Lemma 97.17.1 that $\mathcal{X}$ is an algebraic stack. $\square$

[1] Set theoretical remark: This coproduct is (isomorphic to) an object of $(\mathit{Sch}/S)_{fppf}$ as we have a bound on the index set by axiom [-1], see Sets, Lemma 3.9.9.
[2] The set theoretic condition in Criteria for Representability, Lemma 96.5.5 will hold: the size of the algebraic space $Y$ representing $\mathcal{Y}$ is suitably bounded. Namely, $Y \to S$ will be locally of finite type and $Y$ will satisfy axiom [-1]. Details omitted.

Comments (2)

Comment #2584 by Eric Ahlqvist on

There shouldn't be an "opp" in p : \mathcal{X} \to (\textit{Sch}/S)_{fppf}^{opp} in neither the lemma or the proposition?


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