The Stacks project

94.14.8 Torsors in fppf topology

Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. We define a category $G\textit{-Torsors}$ as follows:

  1. An object of $G\textit{-Torsors}$ is a triple $(U, b, X)$ where

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

    2. $b : U \to B$ is a morphism, and

    3. $X$ is an fppf $G_ U$-torsor over $U$ where $G_ U = U \times _{b, B} G$.

    See Groupoids in Spaces, Definition 77.9.3.

  2. A morphism $(U, b, X) \to (U', b', X')$ is given by a pair $(f, g)$, where $f : U \to U'$ is a morphism of schemes over $B$, and $g : X \to U \times _{f, U'} X'$ is an isomorphism of $G_ U$-torsors.

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

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

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

In the special case $S = B$ the objects are simply pairs $(U, X)$ where $U$ is a scheme over $S$, and $X$ is an fppf $G_ U$-torsor over $U$. Moreover, morphisms are simply cartesian diagrams

\[ \xymatrix{ X \ar[d] \ar[r]_ g & X' \ar[d] \\ U \ar[r]^ f & U' } \]

where $g$ is $G$-equivariant.

Lemma 94.14.9. Up to a replacement as in Stacks, Remark 8.4.9 the functor

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

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, which is Bootstrap, Lemma 79.11.8. We omit the proof of axioms (1) and (2) of Stacks, Definition 8.5.1. $\square$

Lemma 94.14.10. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Denote $\mathcal{G}$, resp. $\mathcal{B}$ the algebraic space $G$, resp. $B$ seen as a sheaf on $(\mathit{Sch}/S)_{fppf}$. The functor

\[ G\textit{-Torsors} \longrightarrow \mathcal{G}/\mathcal{B}\textit{-Torsors} \]

which associates to a triple $(U, b, X)$ the triple $(U, b, \mathcal{X})$ where $\mathcal{X}$ is $X$ viewed as a sheaf is an equivalence of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$.

Proof. We will use the result of Stacks, Lemma 8.4.8 to prove this. The functor is fully faithful since the category of algebraic spaces over $S$ is a full subcategory of the category of sheaves on $(\mathit{Sch}/S)_{fppf}$. Moreover, all objects (on both sides) are locally trivial torsors so condition (2) of the lemma referenced above holds. Hence the functor is an equivalence. $\square$


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