Continuous cohomology group
Definition
Suppose and are topological groups (typically, taken to be T0 topological groups) such that is an abelian group and there is a topological group action of on , i.e., a homomorphism such that the induced map is continuous. We can then define continuous cohomology groups , each of which is an abelian group.
When is understood from context, the subscript may be omitted in the notation for the cohomology group, as well as the notation for the groups of -cocycles and -coboundaries.
Definition in terms of homological algebra
Note that these definitions are the same as those for the cohomology group in the discrete setting, with the following important difference: the -modules here are abelian topological groups with topological group actions of on them, and the morphisms between them must be continuous in addition to being equivariant under the -action. Also, for the bar resolution, the cochains in the cochain complex comprise only the continuous functions.
No. | Shorthand | Detailed description of , the cohomology group |
---|---|---|
1 | Complex based on arbitrary projective resolution | Let be a projective resolution for as a -module with the trivial action. Let be the complex . The cohomology group is defined as the cohomology group for this complex. |
2 | Complex based on arbitrary injective resolution (works if category of -modules has enough injectives!) | Let be an injective resolution for as a -module with the specified action . Let be the complex where has the structure of a trivial action -module. The cohomology group is defined as the cohomology group for this complex. |
3 | As an functor | where is a trivial -module and has the module structure specified by . |
4 | As a right derived functor | , i.e., it is the right derived functor of the invariants functor for (denoted ) evaluated at . The invariants functor sends a -module to its submodule of elements fixed by all elements of . |
5 | Explicit, using the bar resolution | , is defined as the quotient where is the group of cocycles for the action and is the group of coboundaries. |
5' | Explicit, using the normalized bar resolution | Same as definition (5), but we use normalized cocycles and normalized coboundaries instead of arbitrary cocycles and coboundaries. |