Homology group for trivial group action
From Groupprops
Definition
Let be a group and be an abelian group.
The homology groups for trivial group action , also denoted () are abelian groups defined in the following equivalent ways.
Definition in terms of classifying space
can be defined as the homology group , where is the classifying space of and theohomology group is understood to be in the topological sense (singular homology or cellular homology, or any of the equivalent homology theories satisfying the axioms).
Definition as homology group for an action taken as the trivial action
The homology groups for trivial group action are defined as the homology groups where is the trivial map. In other words, we treat as a -module with trivial action of on (i.e., every element of fixes every element of . We thus also treat as a trivial -module, where is a group ring of over the ring of integers .
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 . |