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

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

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.
$\square$

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: