Other Usage
The concept of a principal homogeneous space can also be globalized as follows. Let X be a "space" (a scheme/manifold/topological space etc.), and let G be a group over X, i.e., a group object in the category of spaces over X. In this case, a (right, say) G-torsor E on X is a space E (of the same type) over X with a (right) G action such that the morphism
given by
is an isomorphism in the appropriate category, and such that E is locally trivial on X, in that E→X acquires a section locally on X. Torsors in this sense correspond to classes in the cohomology group H1(X,G).
When we are in the smooth manifold category, then a G-torsor (for G a Lie group) is then precisely a principal G-bundle as defined above.
Read more about this topic: Principal Homogeneous Space
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