G-Torsor In Algebraic Geometry

Definition

Let \(X\) be a topological space (i.e. a scheme what else would it be) and let \(\mathcal G\) be a sheaf of groups, not necessarily commutative. A \(\mathcal G\)-torsor on \(X\) is a sheaf of sets \(\mathcal F\) on \(X\) endowed with an action \(\mathcal G\times \mathcal F\to \mathcal F\) such that

• whenever \(\mathcal F(U)\) is non-empty the action \(\mathcal G(U)\times \mathcal F(U)\to \mathcal F(U)\) is simply transitive
• for every \(x\in X\) the stalk \(\mathcal F_x\) is nonempty.
• \(\mathcal F\)

A morphism of \(\mathcal G\)-torsors \(\mathcal F\to \mathcal F'\) is simply a morphism of sheaves of sets compatible with the \(\mathcal G\)-actions. A trivial \(\mathcal G\)-torsor is one isomorphic to \(\mathcal G\) endowed with its left action on itself.