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The Stacks project

Proposition 101.28.11. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. The following are equivalent

  1. \mathcal{X} is a gerbe over \mathcal{Y}, and

  2. f : \mathcal{X} \to \mathcal{Y} and \Delta : \mathcal{X} \to \mathcal{X} \times _\mathcal {Y} \mathcal{X} are surjective, flat, and locally of finite presentation.

Proof. The implication (1) \Rightarrow (2) follows from Lemmas 101.28.8 and 101.28.10.

Assume (2). It suffices to prove (1) for the base change of f by a surjective, flat, and locally finitely presented morphism \mathcal{Y}' \to \mathcal{Y}, see Lemma 101.28.5 (note that the base change of the diagonal of f is the diagonal of the base change). Thus we may assume \mathcal{Y} is a scheme Y. In this case \mathcal{I}_\mathcal {X} \to \mathcal{X} is a base change of \Delta and we conclude that \mathcal{X} is a gerbe by Proposition 101.28.9. We still have to show that \mathcal{X} is a gerbe over Y. Let \mathcal{X} \to X be the morphism of Lemma 101.28.2 turning \mathcal{X} into a gerbe over the algebraic space X classifying isomorphism classes of objects of \mathcal{X}. It is clear that f : \mathcal{X} \to Y factors as \mathcal{X} \to X \to Y. Since f is surjective, flat, and locally of finite presentation, we conclude that X \to Y is surjective as a map of fppf sheaves (for example use Lemma 101.27.13). On the other hand, X \to Y is injective too: for any scheme T and any two T-valued points x_1, x_2 of X which map to the same point of Y, we can first fppf locally on T lift x_1, x_2 to objects \xi _1, \xi _2 of \mathcal{X} over T and second deduce that \xi _1 and \xi _2 are fppf locally isomorphic by our assumption that \Delta : \mathcal{X} \to \mathcal{X} \times _ Y \mathcal{X} is surjective, flat, and locally of finite presentation. Whence x_1 = x_2 by construction of X. Thus X = Y and the proof is complete. \square


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