39.11 Principal homogeneous spaces
In Cohomology on Sites, Definition 21.4.1 we have defined a torsor for a sheaf of groups on a site. Suppose \tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} is a topology and (G, m) is a group scheme over S. Since \tau is stronger than the canonical topology (see Descent, Lemma 35.13.7) we see that \underline{G} (see Sites, Definition 7.12.3) is a sheaf of groups on (\mathit{Sch}/S)_\tau . Hence we already know what it means to have a torsor for \underline{G} on (\mathit{Sch}/S)_\tau . A special situation arises if this sheaf is representable. In the following definitions we define directly what it means for the representing scheme to be a G-torsor.
Definition 39.11.1. Let S be a scheme. Let (G, m) be a group scheme over S. Let X be a scheme over S, and let a : G \times _ S X \to X be an action of G on X.
We say X is a pseudo G-torsor or that X is formally principally homogeneous under G if the induced morphism of schemes G \times _ S X \to X \times _ S X, (g, x) \mapsto (a(g, x), x) is an isomorphism of schemes over S.
A pseudo G-torsor X is called trivial if there exists an G-equivariant isomorphism G \to X over S where G acts on G by left multiplication.
It is clear that if S' \to S is a morphism of schemes then the pullback X_{S'} of a pseudo G-torsor over S is a pseudo G_{S'}-torsor over S'.
Lemma 39.11.2. In the situation of Definition 39.11.1.
The scheme X is a pseudo G-torsor if and only if for every scheme T over S the set X(T) is either empty or the action of the group G(T) on X(T) is simply transitive.
A pseudo G-torsor X is trivial if and only if the morphism X \to S has a section.
Proof.
Omitted.
\square
Definition 39.11.3. Let S be a scheme. Let (G, m) be a group scheme over S. Let X be a pseudo G-torsor over S.
We say X is a principal homogeneous space or a G-torsor if there exists a fpqc covering1 \{ S_ i \to S\} _{i \in I} such that each X_{S_ i} \to S_ i has a section (i.e., is a trivial pseudo G_{S_ i}-torsor).
Let \tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} . We say X is a G-torsor in the \tau topology, or a \tau G-torsor, or simply a \tau torsor if there exists a \tau covering \{ S_ i \to S\} _{i \in I} such that each X_{S_ i} \to S_ i has a section.
If X is a G-torsor, then we say that it is quasi-isotrivial if it is a torsor for the étale topology.
If X is a G-torsor, then we say that it is locally trivial if it is a torsor for the Zariski topology.
We sometimes say “let X be a G-torsor over S” to indicate that X is a scheme over S equipped with an action of G which turns it into a principal homogeneous space over S. Next we show that this agrees with the notation introduced earlier when both apply.
Lemma 39.11.4. Let S be a scheme. Let (G, m) be a group scheme over S. Let X be a scheme over S, and let a : G \times _ S X \to X be an action of G on X. Let \tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} . Then X is a G-torsor in the \tau -topology if and only if \underline{X} is a \underline{G}-torsor on (\mathit{Sch}/S)_\tau .
Proof.
Omitted.
\square
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