The Stacks project

Lemma 38.37.13. Let $p : \mathcal{S} \to (\mathit{Sch}/S)_ h$ be a category fibred in groupoids. Then $\mathcal{S}$ is a stack in groupoids if and only if the following conditions are satisfied

  1. $\mathcal{S}$ is a stack in groupoids for the Zariski topology,

  2. given a morphism $f : X \to Y$ of $(\mathit{Sch}/S)_ h$ with $Y$ affine and $f$ surjective, flat, proper, and of finite presentation, then

    \[ \mathcal{S}_ Y \longrightarrow \mathcal{S}_ X \times _{\mathcal{S}_{X \times _ Y X}} \mathcal{S}_ X \]

    is an equivalence of categories,

  3. for an almost blow up square as in Example 38.37.10 or 38.37.11 in the category $(\mathit{Sch}/S)_ h$ the functor

    \[ \mathcal{S}_ X \longrightarrow \mathcal{S}_ Z \times _{\mathcal{S}_ E} \mathcal{S}_{X'} \]

    is an equivalence of categories.

Proof. This lemma is a formal consequence of Lemma 38.37.12 and our defnition of stacks in groupoids. For example, assume (1), (2), (3). To show that $\mathcal{S}$ is a stack, we have to prove descent for morphisms and objects, see Stacks, Definition 8.5.1.

If $x, y$ are objects of $\mathcal{S}$ over an object $U$ of $(\mathit{Sch}/S)_ h$, then our assumptions imply $\mathit{Isom}(x, y)$ is a presheaf on $(\mathit{Sch}/U)_ h$ which satisfies (1), (2), (3), and (4) of Lemma 38.37.12 and therefore is a sheaf. Some details omitted.

Let $\{ U_ i \to U\} _{i \in I}$ be a covering of $(\mathit{Sch}/S)_ h$. Let $(x_ i, \varphi _{ij})$ be a descent datum in $\mathcal{S}$ relative to the family $\{ U_ i \to U\} _{i \in I}$, see Stacks, Definition 8.3.1. Consider the rule $F$ which to $V/U$ in $(\mathit{Sch}/U)_ h$ associates the set of pairs $(y, \psi _ i)$ where $y$ is an object of $\mathcal{S}_ V$ and $\psi _ i : y|_{U_ i \times _ U V} \to x_ i|_{U_ i \times _ U V}$ is a morphism of $\mathcal{S}$ over $U_ i \times _ U V$ such that

\[ \varphi _{ij}|_{U_ i \times _ U U_ j \times _ U V} \circ \psi _ i|_{U_ i \times _ U U_ j \times _ U V} = \psi _ j|_{U_ i \times _ U U_ j \times _ U V} \]

up to isomorphism. Since we already have descent for morphisms, it is clear that $F(V/U)$ is either empty or a singleton set. On the other hand, we have $F(U_{i_0}/U)$ is nonempty because it contains $(x_{i_0}, \varphi _{i_0i})$. Since our goal is to prove that $F(U/U)$ is nonempty, it suffices to show that $F$ is a sheaf on $(\mathit{Sch}/U)_ h$. To do this we may use the criterion of Lemma 38.37.12. However, our assumptions (1), (2), (3) imply (by drawing some commutative diagrams which we omit), that properties (1), (2), (3), and (4) of Lemma 38.37.12 hold for $F$.

We omit the verification that if $\mathcal{S}$ is a stack in groupoids, then (1), (2), and (3) are satisfied. $\square$

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