Probably the easiest example of a torsor to understand is the trivial torsor in the topological case.

Definition

Given a space $B$ and a sheaf of groups, $G$ on $B$, the sheaf of sets underlying $G$ has a natural left action by $G$, which is a sheaf morphism. This is transitive etc. and so gives a torsor, called the trivial $G$-torsor, denoted ${T}_{G}$.

It is very important to note that ${T}_{G}$ has ${T}_{G}(B)$ non-empty (i.e., ${T}_{G}$ has a ‘global section’), since it is a group so must have an identity element. Conversely any $G$-torsor which has such a ‘global section’ is isomorphic to ${T}_{G}$.

Revised on September 11, 2010 07:07:33
by Tim Porter
(95.147.237.245)