# Continuous cohomology group

## 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.