78.9 Principal homogeneous spaces
This section is the analogue of Groupoids, Section 39.11. We suggest reading that section first.
Definition 78.9.1. Let S be a scheme. Let B be an algebraic space over S. Let (G, m) be a group algebraic space over B. Let X be an algebraic space over B, and let a : G \times _ B 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 G \times _ B X \to X \times _ B X, (g, x) \mapsto (a(g, x), x) is an isomorphism.
A pseudo G-torsor X is called trivial if there exists an G-equivariant isomorphism G \to X over B where G acts on G by left multiplication.
It is clear that if B' \to B is a morphism of algebraic spaces then the pullback X_{B'} of a pseudo G-torsor over B is a pseudo G_{B'}-torsor over B'.
Lemma 78.9.2. In the situation of Definition 78.9.1.
The algebraic space X is a pseudo G-torsor if and only if for every scheme T over B 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 B has a section.
Proof.
Omitted.
\square
Definition 78.9.3. Let S be a scheme. Let B be an algebraic space over S. Let (G, m) be a group algebraic space over B. Let X be a pseudo G-torsor over B.
We say X is a principal homogeneous space, or more precisely a principal homogeneous G-space over B if there exists a fpqc covering1 \{ B_ i \to B\} _{i \in I} such that each X_{B_ i} \to B_ i has a section (i.e., is a trivial pseudo G_{B_ 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 \{ B_ i \to B\} _{i \in I} such that each X_{B_ i} \to B_ i has a section.
If X is a principal homogeneous G-space over B, then we say that it is quasi-isotrivial if it is a torsor for the étale topology.
If X is a principal homogeneous G-space over B, 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-principal homogeneous space over B” to indicate that X is an algebraic space over B equipped with an action of G which turns it into a principal homogeneous space over B. Next we show that this agrees with the notation introduced earlier when both apply.
Lemma 78.9.4. Let S be a scheme. Let (G, m) be a group algebraic space over S. Let X be an algebraic space over S, and let a : G \times _ S X \to X be an action of G on X. Then X is a G-torsor in the fppf-topology in the sense of Definition 78.9.3 if and only if X is a G-torsor on (\mathit{Sch}/S)_{fppf} in the sense of Cohomology on Sites, Definition 21.4.1.
Proof.
Omitted.
\square
Lemma 78.9.5. Let S be a scheme. Let B be an algebraic space over S. Let G be a group algebraic space over B. Let X be a pseudo G-torsor over B. Assume G and X locally of finite type over B.
If G \to B is unramified, then X \to B is unramified.
If G \to B is locally quasi-finite, then X \to B is locally quasi-finite.
Proof.
Proof of (1). By Morphisms of Spaces, Lemma 67.38.10 we reduce to the case where B is the spectrum of a field. If X is empty, then the result holds. If X is nonempty, then after increasing the field, we may assume X has a point. Then G \cong X and the result holds.
The proof of (2) works in exactly the same way using Morphisms of Spaces, Lemma 67.27.2.
\square
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