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torsor

## Definitions

- A \(G{\hbox{-}}\)torsor is a set with a free transitive \(G{\hbox{-}}\)action. For example, the fibers of a principal bundle are torsors. Given any two torsors, we can compare them using elements of \(G\), but there is no distinguished element. For example, \({\mathbf{A}}_n\) is a torsor over the vector space \(k^n\).

As a birational invariant and relation to being unramified:

Computation

For elliptic curves

In etale cohomology

Relation to central simple algebras

Examples

Something related to the del Pezzo surface and blowup:

Torsors of algebraic groups:

For elliptic curves:

Covering of affine curves:

Covering of varieties:

Principal bundles

See fiber bundle: