Torsor condition

Definition

For $P=\operatorname{Spec}K\to X=\operatorname{Spec}F$, two points of $P$ lying over the same point of $X$ should differ by one and only one Galois automorphism. The torsor condition packages that sentence without choosing geometric points.

Let $G\to X$ be a group scheme and let $P\to X$ carry a right $G$-action. The torsor condition is that $P\to X$ is a cover in the chosen topology and the morphism

is an isomorphism. Thus an ordered pair in one fiber is uniquely a first point together with the group element carrying it to the second. The repeated products are fiber products of schemes.

After a covering base change $U\to X$ admitting a section of $P_U\to U$, the chosen section supplies an equivariant isomorphism