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)