Torsor

Summary

A torsor, or principal homogeneous space, for a group $G$ is a space with a group action of $G$ that is free and transitive. It looks exactly like $G$ except that no point is singled out as the identity. Let $G$ be an algebraic group over a field $k$. A $G$-torsor may have no point over $k$ even though it becomes isomorphic to $G$ over a larger field. Such torsors are classified by the Galois cohomology set $H^1(k, G)$, at least when $G$ is smooth.