Continuous cohomology group

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Definition

Suppose G and A are topological groups (typically, taken to be T0 topological groups) such that A is an abelian group and there is a topological group action of G on A, i.e., a homomorphism \varphi:G \to \operatorname{Aut}(A) such that the induced map G \times A \to A is continuous. We can then define continuous cohomology groups H^n_\varphi(G;A), each of which is an abelian group.

When \varphi is understood from context, the subscript {}_\varphi may be omitted in the notation for the cohomology group, as well as the notation for the groups of n-cocycles and n-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 \mathbb{Z}G-modules here are abelian topological groups with topological group actions of G on them, and the morphisms between them must be continuous in addition to being equivariant under the G-action. Also, for the bar resolution, the cochains in the cochain complex comprise only the continuous functions.

No. Shorthand Detailed description of H^n_\varphi(G,A), the n^{th} cohomology group
1 Complex based on arbitrary projective resolution Let \mathcal{F} be a projective resolution for \mathbb{Z} as a \mathbb{Z}G-module with the trivial action. Let \mathcal{C} be the complex \operatorname{Hom}_{\mathbb{Z}G}(\mathcal{F},A). The cohomology group H^n_\varphi(G,A) is defined as the n^{th} cohomology group for this complex.
2 Complex based on arbitrary injective resolution (works if category of \mathbb{Z}G-modules has enough injectives!) Let \mathcal{I} be an injective resolution for A as a \mathbb{Z}G-module with the specified action \varphi. Let \mathcal{D} be the complex \operatorname{Hom}_{\mathbb{Z}G}(\mathbb{Z},\mathcal{I}) where \mathbb{Z} has the structure of a trivial action \mathbb{Z}G-module. The cohomology group H^n_\varphi(G,A) is defined as the n^{th} cohomology group for this complex.
3 As an \operatorname{Ext} functor \operatorname{Ext}^n_{\mathbb{Z}G}(\mathbb{Z},A) where \mathbb{Z} is a trivial \mathbb{Z}G-module and A has the module structure specified by \varphi.
4 As a right derived functor H^n_\varphi(G,A) = R^n(-^G)(A), i.e., it is the n^{th} right derived functor of the invariants functor for G (denoted -^G) evaluated at A. The invariants functor sends a \mathbb{Z}G-module to its submodule of elements fixed by all elements of G.
5 Explicit, using the bar resolution H^n_\varphi(G,A), is defined as the quotient Z^n_\varphi(G,A)/B^n_\varphi(G,A) where Z^n_\varphi(G,A) is the group of cocycles for the action and B^n_\varphi(G,A) 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.