## 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], , , ,  (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 ). 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], , , , , , and  (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 étale topology by Stacks, Lemmas 8.6.3 and 8.6.7, i.e., axiom  holds. Also $Y$ is limit preserving by Lemma 97.11.2, i.e., we have . Note that $Y$ has (RS), i.e., axiom  holds, by Lemmas 97.5.2 and 97.5.3. Axiom  for $Y$ follows from Lemmas 97.8.1 and 97.8.2. Axiom  follows from Lemmas 97.9.5 and 97.9.6. Axiom  for $Y$ follows directly from openness of versality for $\Delta _\mathcal {X}$ which is part of axiom  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. $\square$

 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.
 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.

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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