Torsor condition - Knowlpedia

Let $G\to X$ be a group scheme and let $P\to X$ carry a right $G$ -action. The torsor condition is that $P\to X$ is a cover in the chosen topology and the morphism

P\times_X G\longrightarrow P\times_X P, \qquad (p,g)\longmapsto(p,p\cdot g)

is an isomorphism. Thus an ordered pair in one fiber is uniquely a first point together with the group element carrying it to the second. The repeated products are fiber products of schemes.

Local triviality

After a covering base change $U\to X$ admitting a section of $P_U\to U$ , the chosen section supplies an equivariant isomorphism

P_U\cong U\times_X G.

Remarks

Checking that an action is free and transitive merely on ordinary topological points is generally too weak. Scheme structure, residue fields, and nilpotents are detected by the displayed isomorphism but may be invisible on the underlying point set.