Principal homogeneous space

In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group G is a non-empty set X on which G acts freely and transitively, meaning that for any x, y in X there exists a unique g in G such that x·g = y where · denotes the (right) action of G on X. An analogous definition holds in other categories where, for example,

If G is nonabelian then one must distinguish between left and right torsors according to whether the action is on the left or right. In this article, we will use right actions. To state the definition more explicitly, X is a G-torsor if X is nonempty and is equipped with a map (in the appropriate category) X × G → X such that

Torsor (algebraic geometry)

In algebraic geometry, given a smooth algebraic group G, a G-torsor or a principal G-bundle P over a scheme X is a scheme (or even algebraic space) with the action of G that is locally trivial in the given Grothendieck topology in the sense that the base change along "some" covering map is the trivial torsor (G acts only on the second factor). Equivalently, a G-torsor P on X is a principal homogeneous space for the group scheme (i.e., acts simply transitively on .)

The definition may be formulated in the sheaf-theoretic language: a sheaf P on the category of X-schemes with some Grothendieck topology is a G-torsor if there is a covering in the topology, called the local trivialization, such that the restriction of P to each is a trivial -torsor.

A line bundle is nothing but a -bundle, and, like a line bundle, the two points of views of torsors, geometric and sheaf-theoretic, are used interchangeably (by permitting P to be a stack like an algebraic space if necessary).