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 $(\mathit{Sch}/S)_{fppf}$.
95.14 Classifying torsors
We want to carefuly explain a number of variants of what it could mean to study the stack of torsors for a group algebraic space $G$ or a sheaf of groups $\mathcal{G}$.
95.14.1 Torsors for a sheaf of groups
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.
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. $\square$
95.14.3 Variant on torsors for a sheaf
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 defines a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$.
\[ p : \mathcal{G}/\mathcal{B}\textit{-Torsors} \longrightarrow (\mathit{Sch}/S)_{fppf} \]
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. $\square$
95.14.5 Principal homogeneous spaces
Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. We define a category $G\textit{-Principal}$ as follows:
An object of $G\textit{-Principal}$ is a triple $(U, b, X)$ where
$U$ is an object of $(\mathit{Sch}/S)_{fppf}$,
$b : U \to B$ is a morphism over $S$, and
$X$ is a principal homogeneous $G_ U$-space over $U$ where $G_ U = U \times _{b, B} G$.
See Groupoids in Spaces, Definition 78.9.3.
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 principal homogeneous $G_ U$-spaces.
Thus $G\textit{-Principal}$ is a category and
\[ p : G\textit{-Principal} \longrightarrow (\mathit{Sch}/S)_{fppf}, \quad (U, b, X) \longmapsto U \]
is a functor. Note that the fibre category of $G\textit{-Principal}$ over $U$ is the disjoint union over $b : U \to B$ of the categories of principal homogeneous $U \times _{b, B} G$-spaces, 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 a principal homogeneous $G_ U$-space 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.
Remark 95.14.6. We conjecture that up to a replacement as in Stacks, Remark 8.4.9 the functor defines a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$. This would follow if one could show that given
a covering $\{ U_ i \to U\} _{i \in I}$ of $(\mathit{Sch}/S)_{fppf}$,
an group algebraic space $H$ over $U$,
for every $i$ a principal homogeneous $H_{U_ i}$-space $X_ i$ over $U_ i$, and
$H$-equivariant isomorphisms $\varphi _{ij} : X_{i, U_ i \times _ U U_ j} \to X_{j, U_ i \times _ U U_ j}$ satisfying the cocycle condition,
\[ p : G\textit{-Principal} \longrightarrow (\mathit{Sch}/S)_{fppf} \]
there exists a principal homogeneous $H$-space $X$ over $U$ which recovers $(X_ i, \varphi _{ij})$. The technique of the proof of Bootstrap, Lemma 80.11.8 reduces this to a set theoretical question, so the reader who ignores set theoretical questions will “know” that the result is true. In https://math.columbia.edu/~dejong/wordpress/?p=591 there is a suggestion as to how to approach this problem.
95.14.7 Variant on principal homogeneous spaces
Let $S$ be a scheme. Let $B = S$. Let $G$ be a group scheme over $B = S$. In this setting we can define a full subcategory $G\textit{-Principal-Schemes} \subset G\textit{-Principal}$ whose objects are pairs $(U, X)$ where $U$ is an object of $(\mathit{Sch}/S)_{fppf}$ and $X \to U$ is a principal homogeneous $G$-space over $U$ which is representable, i.e., a scheme.
It is in general not the case that $G\textit{-Principal-Schemes}$ is a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$. The reason is that in general there really do exist principal homogeneous spaces which are not schemes, hence descent for objects will not be satisfied in general.
95.14.8 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:
An object of $G\textit{-Torsors}$ is a triple $(U, b, X)$ where
$U$ is an object of $(\mathit{Sch}/S)_{fppf}$,
$b : U \to B$ is a morphism, and
$X$ is an fppf $G_ U$-torsor over $U$ where $G_ U = U \times _{b, B} G$.
See Groupoids in Spaces, Definition 78.9.3.
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 95.14.9. Up to a replacement as in Stacks, Remark 8.4.9 the functor defines a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$.
\[ p : G\textit{-Torsors} \longrightarrow (\mathit{Sch}/S)_{fppf} \]
Proof. The most difficult part of the proof is to show that we have descent for objects, which is Bootstrap, Lemma 80.11.8. We omit the proof of axioms (1) and (2) of Stacks, Definition 8.5.1. $\square$
Lemma 95.14.10. 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 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 $(\mathit{Sch}/S)_{fppf}$.
\[ G\textit{-Torsors} \longrightarrow \mathcal{G}/\mathcal{B}\textit{-Torsors} \]
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. $\square$
95.14.11 Variant on torsors in fppf topology
Let $S$ be a scheme. Let $B = S$. Let $G$ be a group scheme over $B = S$. In this setting we can define a full subcategory $G\textit{-Torsors-Schemes} \subset G\textit{-Torsors}$ whose objects are pairs $(U, X)$ where $U$ is an object of $(\mathit{Sch}/S)_{fppf}$ and $X \to U$ is an fppf $G$-torsor over $U$ which is representable, i.e., a scheme.
It is in general not the case that $G\textit{-Torsors-Schemes}$ is a stack in groupoids over $(\mathit{Sch}/S)_{fppf}$. The reason is that in general there really do exist fppf $G$-torsors which are not schemes, hence descent for objects will not be satisfied in general.
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