## 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.

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