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 an 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 .) It is common to consider a torsor for not just a group scheme but more generally for a group sheaf (e.g., fppf group sheaf).