Lemma 95.14.4. Up to a replacement as in Stacks, Remark 8.4.9 the functor

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

The construction of Subsection 95.14.1 can be generalized slightly. Namely, let $\mathcal{G} \to \mathcal{B}$ be a map of sheaves on $(\mathit{Sch}/S)_{fppf}$ and let

\[ m : \mathcal{G} \times _\mathcal {B} \mathcal{G} \longrightarrow \mathcal{G} \]

be a group law on $\mathcal{G}/\mathcal{B}$. In other words, the pair $(\mathcal{G}, m)$ is a group object of the topos $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})/\mathcal{B}$. See Sites, Section 7.30 for information regarding localizations of topoi. In this setting we can define a category $\mathcal{G}/\mathcal{B}\textit{-Torsors}$ as follows (where we use the Yoneda embedding to think of schemes as sheaves):

An object of $\mathcal{G}/\mathcal{B}\textit{-Torsors}$ is a triple $(U, b, \mathcal{F})$ where

$U$ is an object of $(\mathit{Sch}/S)_{fppf}$,

$b : U \to \mathcal{B}$ is a section of $\mathcal{B}$ over $U$, and

$\mathcal{F}$ is a $U \times _{b, \mathcal{B}}\mathcal{G}$-torsor over $U$.

A morphism $(U, b, \mathcal{F}) \to (U', b', \mathcal{F}')$ is given by a pair $(f, g)$, where $f : U \to U'$ is a morphism of schemes over $S$ such that $b = b' \circ f$, and $g : f^{-1}\mathcal{F}' \to \mathcal{F}$ is an isomorphism of $U \times _{b, \mathcal{B}} \mathcal{G}$-torsors.

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

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

is a functor. Note that the fibre category of $\mathcal{G}/\mathcal{B}\textit{-Torsors}$ over $U$ is the disjoint union over $b : U \to \mathcal{B}$ of the categories of $U \times _{b, \mathcal{B}} \mathcal{G}$-torsors, hence is a groupoid.

In the special case $\mathcal{B} = S$ we recover the category $\mathcal{G}\textit{-Torsors}$ introduced in Subsection 95.14.1.

Lemma 95.14.4. Up to a replacement as in Stacks, Remark 8.4.9 the functor

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

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

**Proof.**
This proof is a repeat of the proof of Lemma 95.14.2. The reader is encouraged to read that proof first since the notation is less cumbersome. 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 pair $(b_ i, \mathcal{F}_ i)$ consisting of a morphism $b_ i : U_ i \to \mathcal{B}$ and a $U_ i \times _{b_ i, \mathcal{B}} \mathcal{G}$-torsor $\mathcal{F}_ i$, and for each $i, j \in I$ we have $b_ i|_{U_ i \times _ U U_ j} = b_ j|_{U_ i \times _ U U_ j}$ and we are given an isomorphism $\varphi _{ij} : \mathcal{F}_ i|_{U_ i \times _ U U_ j} \to \mathcal{F}_ j|_{U_ i \times _ U U_ j}$ of $(U_ i \times _ U U_ j) \times _\mathcal {B} \mathcal{G}$-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 sheaf axiom for $\mathcal{B}$ the morphisms $b_ i$ come from a unique morphism $b : U \to \mathcal{B}$. By the equivalence of categories in Sites, Lemma 7.26.5 the action maps $(U_ i \times _{b_ i, \mathcal{B}} \mathcal{G}) \times _{U_ i} \mathcal{F}_ i \to \mathcal{F}_ i$ glue to give a map $(U \times _{b, \mathcal{B}} \mathcal{G}) \times \mathcal{F} \to \mathcal{F}$. Now we have to show that this is an action and that $\mathcal{F}$ becomes a $U \times _{b, \mathcal{B}} \mathcal{G}$-torsor. Both properties may be checked locally, and hence follow from the corresponding properties of the actions on the $\mathcal{F}_ i$. This proves that descent for objects holds in $\mathcal{G}/\mathcal{B}\textit{-Torsors}$. Some details omitted.
$\square$

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)