The Stacks project

Lemma 94.14.2. Up to a replacement as in Stacks, Remark 8.4.9 the functor

\[ p : \mathcal{G}\textit{-Torsors} \longrightarrow (\mathit{Sch}/S)_{fppf} \]

defines a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$.

Proof. The most difficult part of the proof is to show that we have descent for objects. Let $\{ U_ i \to U\} _{i \in I}$ be a covering of $(\mathit{Sch}/S)_{fppf}$. Suppose that for each $i$ we are given a $\mathcal{G}|_{U_ i}$-torsor $\mathcal{F}_ i$, and for each $i, j \in I$ an isomorphism $\varphi _{ij} : \mathcal{F}_ i|_{U_ i \times _ U U_ j} \to \mathcal{F}_ j|_{U_ i \times _ U U_ j}$ of $\mathcal{G}|_{U_ i \times _ U U_ j}$-torsors satisfying a suitable cocycle condition on $U_ i \times _ U U_ j \times _ U U_ k$. Then by Sites, Section 7.26 we obtain a sheaf $\mathcal{F}$ on $(\mathit{Sch}/U)_{fppf}$ whose restriction to each $U_ i$ recovers $\mathcal{F}_ i$ as well as recovering the descent data. By the equivalence of categories in Sites, Lemma 7.26.5 the action maps $\mathcal{G}|_{U_ i} \times \mathcal{F}_ i \to \mathcal{F}_ i$ glue to give a map $a : \mathcal{G}|_ U \times \mathcal{F} \to \mathcal{F}$. Now we have to show that $a$ is an action and that $\mathcal{F}$ becomes a $\mathcal{G}|_ U$-torsor. Both properties may be checked locally, and hence follow from the corresponding properties of the actions $\mathcal{G}|_{U_ i} \times \mathcal{F}_ i \to \mathcal{F}_ i$. This proves that descent for objects holds in $\mathcal{G}\textit{-Torsors}$. Some details omitted. $\square$

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