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
20120924