77.9 Principal homogeneous spaces

This section is the analogue of Groupoids, Section 39.11. We suggest reading that section first.

Definition 77.9.1. 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 $X$.

1. 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.

2. 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 77.9.2. In the situation of Definition 77.9.1.

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.

2. A pseudo $G$-torsor $X$ is trivial if and only if the morphism $X \to B$ has a section.

Proof. Omitted. $\square$

Definition 77.9.3. 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 $B$.

1. 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).

2. 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.

3. 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.

4. 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 77.9.4. 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 77.9.3 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 77.9.5. 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 $B$.

1. If $G \to B$ is unramified, then $X \to B$ is unramified.

2. 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 66.38.10 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 66.27.2. $\square$

[1] The default type of torsor in Groupoids, Definition 39.11.3 is a pseudo torsor which is trivial on an fpqc covering. Since $G$, as an algebraic space, can be seen a sheaf of groups there already is a notion of a $G$-torsor which corresponds to fppf-torsor, see Lemma 77.9.4. Hence we use “principal homogeneous space” for a pseudo torsor which is fpqc locally trivial, and we try to avoid using the word torsor in this situation.

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).