group cohomology

Let $G$ be a group and let $M$ be a (left) $G$-module. The$0^{th}$ cohomology group of the $G$-module $M$ is

$$H^0(G,M)=\{m\in M:\forall \sigma \in G,\sigma m=m\}$$

which is the set of elements of $M$ which are $G$-invariant, alsodenoted by $M^G$.

A map $\varphi :G\to M$ is said to be a crossedhomomorphism (or 1-cocycle) if

$$\varphi (\alpha \beta )=\varphi (\alpha )+\alpha \varphi (\beta )$$

for all $\alpha ,\beta \in G$. If we fix $m\in M$, the map$\rho :G\to M$ defined by

$$\rho (\alpha )=\alpha m-m$$

is clearly a crossed homomorphism, said to be principal (or1-coboundary). We define the following groups:

$Z^1(G,M)$$=$$\mathrm{\{}\varphi :G\to M:\varphi \text{ is a 1-cocycle}\}$

$B^1(G,M)$$=$$\mathrm{\{}\rho :G\to M:\rho \text{ is a1-coboundary}\}$

Finally, the $1^{st}$ cohomology group of the $G$-module$M$ is defined to be the quotient group:

$$H^1(G,M)=Z^1(G,M)/B^1(G,M)$$

The following proposition is very useful when trying to computecohomology groups:

Proposition 1.

Let $G$ be a group and let $A\mathrm{,}B\mathrm{,}C$ be $G$-modules related by anexact sequence:

$$0\to A\to B\to C\to 0$$

Then there is a long exact sequence in cohomology:

$$0\to H^0(G,A)\to H^0(G,B)\to H^0(G,C)\to H^1(G,A)\to H^1(G,B)\to H^1(G,C)\to \mathrm{\dots }$$

In general, the cohomology groups $H^n(G,M)$ can be defined asfollows:

Definition 1.

Define $C^{\mathrm{0}}\mathit{}\mathrm{(}G\mathrm{,}M\mathrm{)}\mathrm{=}M$ and for $n\mathrm{\ge }\mathrm{1}$ define the additive group:

$$C^n(G,M)=\{\varphi :G^n\to M\}$$

The elements of $C^n\mathit{}\mathrm{(}G\mathrm{,}M\mathrm{)}$ are called $n$-cochains. Also, for$n\mathrm{\ge }\mathrm{0}$ define the $n^{t\mathit{}h}$ coboundary homomorphism $d_n\mathrm{:}{C}^n\mathit{}\mathrm{(}G\mathrm{,}M\mathrm{)}\mathrm{\to }{C}^{n\mathrm{+}\mathrm{1}}\mathit{}\mathrm{(}G\mathrm{,}M\mathrm{)}$:

$d_n(\varphi )(g_1,\mathrm{\dots },g_{n+1})$$=$$g_1\cdot \varphi (g_2,\mathrm{\dots },g_{n+1})$

$+$${\displaystyle \sum _{i=1}^n}{(-1)}^i\varphi (g_1,\mathrm{\dots },g_{i-1},g_i{g}_{i+1},g_{i+2},\mathrm{\dots },g_{n+1})$

$+$${(-1)}^{n+1}\varphi (g_1,\mathrm{\dots },g_n)$

Let $Z^n\mathit{}\mathrm{(}G\mathrm{,}M\mathrm{)}\mathrm{=}\ker \mathit{}{d}_n$ for $n\mathrm{\ge }\mathrm{0}$, the set of$n$-cocyles. Also, let $B^{\mathrm{0}}\mathit{}\mathrm{(}G\mathrm{,}M\mathrm{)}\mathrm{=}\mathrm{1}$ and for $n\mathrm{\ge }\mathrm{1}$ let$B^n\mathit{}\mathrm{(}G\mathrm{,}M\mathrm{)}\mathrm{=}\mathrm{image}\mathit{}{d}_{n\mathrm{-}\mathrm{1}}$, the set of$n$-coboundaries.

Finally we define the $n^{t\mathit{}h}$-cohomology group of $G$ withcoefficients in $M$ to be

$$H^n(G,M)=Z^n(G,M)/B^n(G,M)$$