Principal G-bundle on a scheme

Definition

Let $G$ be a group scheme over a base $S$ , and let $X$ be an $S$ -scheme. A principal $G$ -bundle on $X$ is a $G_X$ -torsor on a specified site of $X$ , commonly the fppf or étale site, whose underlying torsor is represented by an $X$ -scheme $P$ . Thus $P\to X$ has a right action of $G_X=G\times_SX$ and is locally $G_X$ -equivariantly isomorphic to $G_X$ , with $G_X$ acting on itself by translation.

The torsor identity is expressed by the isomorphism

P\times_S G \longrightarrow P\times_XP,\qquad (p,g)\longmapsto(p,pg).

Relation to differential geometry

When $S=\operatorname{Spec}\mathbb C$ , $G$ is a complex algebraic group, and $X$ is a complex algebraic variety, analytification produces a holomorphic principal $G(\mathbb C)$ -bundle. Its underlying smooth bundle is a smooth principal bundle, but passing between algebraic, holomorphic, and smooth categories requires comparison results and can lose structure.

Moduli

When $X$ is a fixed curve, allowing $P$ to vary in families over test schemes gives the moduli stack of $G$ -bundles.

References

Alexander Grothendieck, “Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats,” Séminaire Bourbaki 190 (1959–1960).