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trivial torsor

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

Definition

Given a space BB and a sheaf of groups, GG on BB, the sheaf of sets underlying GG has a natural left action by GG, which is a sheaf morphism. This is transitive etc. and so gives a torsor, called the trivial GG-torsor, denoted T GT_G.

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

Last revised on September 11, 2010 at 07:07:33. See the history of this page for a list of all contributions to it.