## 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 35.10.7) 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 covering^{1} $\{ S_ i \to S\} _{i \in I}$ 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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