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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 with an action of G that is locally trivial in the given Grothendieck topology in the sense that the base change Y × X P {\displaystyle Y\times _{X}P} along "some" covering map Y → X {\displaystyle Y\to X} is the trivial torsor Y × G → Y {\displaystyle Y\times G\to Y} . Equivalently, a G-torsor P on X is a principal homogeneous space for the group scheme G X = X × G {\displaystyle G_{X}=X\times G}
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 i → X } {\displaystyle \{U_{i}\to X\}} in the topology, called the local trivialization, such that the restriction of P to each U i {\displaystyle U_{i}} is a trivial G U i {\displaystyle G_{U_{i}}} -torsor.
A line bundle is nothing but a G m {\displaystyle \mathbb {G} _{m}} -bundle, and, like a line bundle, the two points of views of torsors, geometric and sheaf-theoretic, are used interchangeably.
It is common to consider a torsor for not just a group scheme but more generally for a group sheaf.