## 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 $X$.

1. 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 $S$.

2. 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.

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.

2. 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 $S$.

1. 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).

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 $\{ S_ i \to S\} _{i \in I}$ such that each $X_{S_ i} \to S_ i$ has a section.

3. If $X$ is a $G$-torsor, then we say that it is quasi-isotrivial if it is a torsor for the étale topology.

4. 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. $\square$

Remark 39.11.5. Let $(G, m)$ be a group scheme over the scheme $S$. In this situation we have the following natural types of questions:

1. If $X \to S$ is a pseudo $G$-torsor and $X \to S$ is surjective, then is $X$ necessarily a $G$-torsor?

2. Is every $\underline{G}$-torsor on $(\mathit{Sch}/S)_{fppf}$ representable? In other words, does every $\underline{G}$-torsor come from a fppf $G$-torsor?

3. Is every $G$-torsor an fppf (resp. smooth, resp. étale, resp. Zariski) torsor?

In general the answers to these questions is no. To get a positive answer we need to impose additional conditions on $G \to S$. For example: If $S$ is the spectrum of a field, then the answer to (1) is yes because then $\{ X \to S\}$ is a fpqc covering trivializing $X$. If $G \to S$ is affine, then the answer to (2) is yes (insert future reference here). If $G = \text{GL}_{n, S}$ then the answer to (3) is yes and in fact any $\text{GL}_{n, S}$-torsor is locally trivial (insert future reference here).

[1] This means that the default type of torsor is a pseudo torsor which is trivial on an fpqc covering. This is the definition in [Exposé IV, 6.5, SGA3]. It is a little bit inconvenient for us as we most often work in the fppf topology.

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