Example Application of the Theorem

For example, let's see what we get from the exact sequence

where is a positive integer, and has the structure of trivial  module. By definition we have and . The long exact sequence begins

From the first few terms of the sequence and the fact that surjects onto , we see that on is injective. This is consistent with our observation above that . Using this vanishing and the right side of the exact sequence we obtain an isomorphism

As we observed above, when a group acts trivially the is , so

 (11.1)

One can prove that for any and any module  that the group has exponent dividing (see Remark 11.3.4). Thus (11.2.1) allows us to understand , and this comprehension arose naturally from the properties that determine .

William Stein 2012-09-24