Homology group
Definition
Let be a group acting on an abelian group
, via an action
. Equivalently,
is a module over the (possibly non-commutative) unital group ring
of
over the ring of integers.
The homology groups (
) are abelian groups defined in the following equivalent ways.
When is understood from context, the subscript
may be omitted in the notation for the homology group, as well as the notation for the groups of
-cycles and
-boundaries.
No. | Shorthand | Detailed description of ![]() ![]() |
---|---|---|
1 | Explicit, using the bar resolution | ![]() ![]() ![]() ![]() |
2 | Complex based on arbitrary projective resolution | Let ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | As a ![]() |
![]() ![]() ![]() ![]() ![]() |
4 | As a left derived functor | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Equivalence of definitions
Further information: Equivalence of definitions of homology group
The equivalence of (1) and (2) follows from the fact that (1) is the special case of (2) that arises if we choose our projective resolution as the bar resolution. The equivalence between (2) and (3) is by the definition of . The equivalence between (3) and (4) follows from the fact that the coinvariants functor (defined in (4)) sends
to
where
is given the structure of a trivial
-module.