Lemma 95.14.2. Up to a replacement as in Stacks, Remark 8.4.9 the functor

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

Let $\mathcal{G}$ be a sheaf of groups on $(\mathit{Sch}/S)_{fppf}$. For $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ we denote $\mathcal{G}|_ U$ the restriction of $\mathcal{G}$ to $(\mathit{Sch}/U)_{fppf}$. We define a category $\mathcal{G}\textit{-Torsors}$ as follows:

An object of $\mathcal{G}\textit{-Torsors}$ is a pair $(U, \mathcal{F})$ where $U$ is an object of $(\mathit{Sch}/S)_{fppf}$ and $\mathcal{F}$ is a $\mathcal{G}|_ U$-torsor, see Cohomology on Sites, Definition 21.4.1.

A morphism $(U, \mathcal{F}) \to (V, \mathcal{H})$ is given by a pair $(f, \alpha )$, where $f : U \to V$ is a morphism of schemes over $S$, and $\alpha : f^{-1}\mathcal{H} \to \mathcal{F}$ is an isomorphism of $\mathcal{G}|_ U$-torsors.

Thus $\mathcal{G}\textit{-Torsors}$ is a category and

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

is a functor. Note that the fibre category of $\mathcal{G}\textit{-Torsors}$ over $U$ is the category of $\mathcal{G}|_ U$-torsors which is a groupoid.

Lemma 95.14.2. 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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