The Stacks project

Lemma 94.15.2. Up to a replacement as in Stacks, Remark 8.4.9 the functor

\[ p : [[X/G]] \longrightarrow (\mathit{Sch}/S)_{fppf} \]

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. Suppose that $\{ U_ i \to U\} _{i \in I}$ is a covering in $(\mathit{Sch}/S)_{fppf}$. Let $\xi _ i = (U_ i, b_ i, P_ i, \varphi _ i)$ be objects of $[[X/G]]$ over $U_ i$, and let $\varphi _{ij} : \text{pr}_0^*\xi _ i \to \text{pr}_1^*\xi _ j$ be a descent datum. This in particular implies that we get a descent datum on the triples $(U_ i, b_ i, P_ i)$ for the stack in groupoids $G\textit{-Torsors}$ by applying the functor $[[X/G]] \to G\textit{-Torsors}$ above. We have seen that $G\textit{-Torsors}$ is a stack in groupoids (Lemma 94.14.9). Hence we may assume that $b_ i = b|_{U_ i}$ for some morphism $b : U \to B$, and that $P_ i = U_ i \times _ U P$ for some fppf $G_ U = U \times _{b, B} G$-torsor $P$ over $U$. The morphisms $\varphi _ i$ are compatible with the canonical descent datum on the restrictions $U_ i \times _ U P$ and hence define a morphism $\varphi : P \to X$. (For example you can use Sites, Lemma 7.26.5 or you can use Descent on Spaces, Lemma 73.7.2 to get $\varphi $.) This proves descent for objects. We omit the proof of axioms (1) and (2) of Stacks, Definition 8.5.1. $\square$


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