trivial torsor

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


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 (