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