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 35.13.7) 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 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).
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.
$\square$
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 (this follows from Descent, Lemma 35.37.1). 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 (this follows from Descent, Lemma 35.7.6).
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