The Stacks project

Lemma 94.18.1. The category $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ endowed with the functor $p$ above defines a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$.

Proof. As usual, the hardest part is to show descent for objects. To see this let $\{ U_ i \to U\} $ be a covering of $(\mathit{Sch}/S)_{fppf}$. Let $\xi _ i = (U_ i, Z_ i, y_ i, x_ i, \alpha _ i)$ be an object of $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$ lying over $U_ i$, and let $\varphi _{ij} : \text{pr}_0^*\xi _ i \to \text{pr}_1^*\xi _ j$ be a descent datum. First, observe that $\varphi _{ij}$ induces a descent datum $(Z_ i/U_ i, \varphi _{ij})$ which is effective by Descent, Lemma 35.37.1 This produces a scheme $Z/U$ which is finite locally free of degree $d$ by Descent, Lemma 35.23.30. From now on we identify $Z_ i$ with $Z \times _ U U_ i$. Next, the objects $y_ i$ in the fibre categories $\mathcal{Y}_{U_ i}$ descend to an object $y$ in $\mathcal{Y}_ U$ because $\mathcal{Y}$ is a stack in groupoids. Similarly the objects $x_ i$ in the fibre categories $\mathcal{X}_{Z_ i}$ descend to an object $x$ in $\mathcal{X}_ Z$ because $\mathcal{X}$ is a stack in groupoids. Finally, the given isomorphisms

\[ \alpha _ i : (y|_ Z)_{Z_ i} = y_ i|_{Z_ i} \longrightarrow F(x_ i) = F(x|_{Z_ i}) \]

glue to a morphism $\alpha : y|_ Z \to F(x)$ as the $\mathcal{Y}$ is a stack and hence $\mathit{Isom}_\mathcal {Y}(y|_ Z, F(x))$ is a sheaf. Details omitted. $\square$

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