Group Cohomology - Core Definitions

Group cohomology provides a powerful tool for studying groups through their module representations.

Definition6.1Group Cohomology

For a group $G$ and $G$ -module $M$ , the group cohomology is defined as: $H^n(G, M) = \text{Ext}^n_{\mathbb{Z}[G]}(\mathbb{Z}, M)$

where $\mathbb{Z}$ has trivial $G$ -action. Equivalently: $H^n(G, M) = R^n(\text{Inv}_G)(M)$

where $\text{Inv}_G(M) = M^G = \{m \in M : gm = m \text{ for all } g \in G\}$ is the invariants functor.

Definition6.2Bar Resolution

The bar resolution provides an explicit free $\mathbb{Z}[G]$ -resolution of $\mathbb{Z}$ : $\cdots \to \mathbb{Z}[G^{n+1}] \xrightarrow{\partial_n} \mathbb{Z}[G^n] \to \cdots \to \mathbb{Z}[G] \to \mathbb{Z} \to 0$

The differential is: $\partial_n(g_0, \ldots, g_n) = \sum_{i=0}^n (-1)^i (g_0, \ldots, \hat{g}_i, \ldots, g_n)$

where $\hat{g}_i$ means omitting $g_i$ .

ExampleLow-Degree Cohomology

For a $G$ -module $M$ :

$H^0(G, M) = M^G$ (invariants)
$H^1(G, M) = \text{Der}(G, M) / \text{PDer}(G, M)$ (derivations mod principal derivations)
$H^2(G, M)$ classifies extensions $1 \to M \to E \to G \to 1$

Definition6.3Group Homology

The group homology is defined dually: $H_n(G, M) = \text{Tor}_n^{\mathbb{Z}[G]}(\mathbb{Z}, M)$

where $\mathbb{Z}$ has trivial $G$ -action. This computes: $H_n(G, M) = H_n(B_\bullet \otimes_{\mathbb{Z}[G]} M)$

using the bar resolution.

ExampleCyclic Groups

For a cyclic group $G = \langle g \rangle$ of order $n$ and $G$ -module $M$ : $H^{2k}(G, M) \cong M^G / N_G M$ $H^{2k+1}(G, M) \cong (M_{G}) / (g-1)M$

where $N_G = 1 + g + g^2 + \cdots + g^{n-1}$ is the norm and $M_G = \{m : N_G m = 0\}$ .

Remark

Group cohomology is periodic for finite cyclic groups, with period 2. This periodicity is a special feature not shared by general groups.