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  "body": "\n## Summary\n\nA torsor, or principal homogeneous space, for a group $G$ is a space with a [[group action|0WCY42EV9V]] of $G$ that is free and transitive.\nIt looks exactly like $G$ except that no point is singled out as the identity.\nLet $G$ be an [[algebraic group|GH81SYAPPN]] over a field $k$.\nA $G$-torsor may have no point over $k$ even though it becomes isomorphic to $G$ over a larger field.\nSuch torsors are classified by the [[Galois cohomology|9MPCZ3EAJM]] set $H^1(k, G)$, at least when $G$ is smooth.\n\n## Explanations\n\n### L1 human\n\nThe points of a line form a torsor for the group of translations.\nTwo points differ by exactly one translation, but there is no natural origin, so one cannot add points, only move them.\nSimilarly, the musical notes form a torsor under shifts by intervals, and calendar dates form a torsor under shifts by days.\nA torsor is a group that has forgotten which element is the identity.\n",
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