Let be the submodule of elements of that are fixed by . Notice that if is a homomorphism of -modules, then restriction defines a homomorphism , so is a functor. In fact, it is a left-exact functor:
We construct explicitly as follows.
Consider as a -module, equipped with the
trivial -action.
Consider the following free resolution of .
Let be the free -module with basis
the set of tuples
, and
with acting on componentwise:
The cohomology groups are then
the cohomology groups of the complex
.
We identify an element of
with a function
such that the condition
The boundary map
on such functions
is then given explicitly by the formula
The group of
-cocycles is the group of
,
as above are functions of variables such that
.
The subgroup of -coboundaries
is the image of
under .
Explicitly, the cohomology group is
the quotient of the group group of -cocycles
modulo the subgroup of -coboundaries.
When , the -cocycles
are the maps such that
William 2007-05-25