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 , the homology group|
|1||Explicit, using the bar resolution||, is defined as the quotient where is the group of cycles for the action and is the group of boundaries.|
|2||Complex based on arbitrary projective resolution||Let be a projective resolution for as a -module with the trivial action. Let be the complex . The homology group is defined as the homology group for this complex.|
|3||As a functor||where is a trivial -module and has the module structure specified by .|
|4||As a left derived functor||, i.e., it is the left derived functor of the coinvariants functor for (denoted ) evaluated at . The coinvariants functor sends a -module to where is generated by all elements of the form .|
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.