What is "torsor"

n. (context mathematics English) A principal homogeneous space for a group, such that the stabilizer subgroup of any point is trivial.

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 Y × P along "some" covering map Y → X is the trivial torsor Y × G → Y (G acts only on the second factor). Equivalently, a G-torsor P on X is a principal homogeneous space for the group scheme G = X × G (i.e., G acts simply transitively on P.)

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 {U → X} in the topology, called the local trivialization, such that the restriction of P to each U is a trivial G-torsor.

A line bundle is nothing but a G-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).