Proposition 98.17.2. Let S be a locally Noetherian scheme. Let p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf} be a category fibred in groupoids. Assume that
\Delta _\Delta : \mathcal{X} \to \mathcal{X} \times _{\mathcal{X} \times \mathcal{X}} \mathcal{X} is representable by algebraic spaces,
\mathcal{X} satisfies axioms [-1], [0], [1], [2], [3], [4], and [5] (see Section 98.14),
\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 97.5.51. 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 98.16.1 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 97.4.4. Observe that Y is a sheaf for the étale topology by Stacks, Lemmas 8.6.3 and 8.6.7, i.e., axiom [0] holds. Also Y is limit preserving by Lemma 98.11.2, i.e., we have [1]. Note that Y has (RS), i.e., axiom [2] holds, by Lemmas 98.5.2 and 98.5.3. Axiom [3] for Y follows from Lemmas 98.8.1 and 98.8.2. Axiom [4] follows from Lemmas 98.9.5 and 98.9.6. 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 98.16.1 are satisfied and Y is an algebraic space.
At this point it follows from Lemma 98.17.1 that \mathcal{X} is an algebraic stack.
\square
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