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 ${\displaystyle Y\times _{X}P}$ along "some" covering map ${\displaystyle Y\to X}$ is the trivial torsor ${\displaystyle Y\times G\to Y}$ (G acts only on the second factor). Equivalently, a G-torsor P on X is a principal homogeneous space for the group scheme ${\displaystyle G_{X}=X\times G}$ (i.e., ${\displaystyle G_{X}}$ acts simply transitively on ${\displaystyle P}$.)

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