Group Cohomology - Applications

Group cohomology has profound applications in number theory, topology, and representation theory.

Theorem6.15Hilbert's Theorem 90 (Additive Form)

For a cyclic Galois extension $L/K$ with group $G = \langle \sigma \rangle$ : $H^1(G, L) = 0$

where $L$ is viewed as a $G$ -module under addition. The multiplicative form states $H^1(G, L^*) = 0$ .

Remark

Hilbert's Theorem 90 is the foundation of Galois cohomology and is essential in algebraic number theory for studying norms and class groups.

Theorem6.16Brauer Group and H²

For a field $K$ and its separable closure $K^s$ with Galois group $G = \text{Gal}(K^s/K)$ : $\text{Br}(K) \cong H^2(G, (K^s)^*)$

The Brauer group classifies central simple algebras over $K$ .

ExampleClass Field Theory

In class field theory, the reciprocity map is described via group cohomology: $H^2(G_K, \mathbb{C}^*) \cong \mathbb{Q}/\mathbb{Z}$

where $G_K$ is the absolute Galois group. This connects to local and global class field theory.

Theorem6.17Group Extensions and H²

There is a bijection: $H^2(G, M) \cong \{\text{central extensions } 1 \to M \to E \to G \to 1\} / \sim$

where $M$ is abelian and in the center of $E$ . This generalizes to non-central extensions via $H^2$ with non-abelian coefficients.

ExampleSchur Multiplier

The Schur multiplier of $G$ is $H^2(G, \mathbb{C}^*)$ , which governs:

Central extensions of $G$
Projective representations of $G$
Universal central extensions

For finite simple groups, the Schur multiplier has been completely classified.

Theorem6.18Cohomology of Profinite Groups

For a profinite group $G$ (e.g., Galois groups), continuous cohomology is defined using continuous cochains. This gives: $H^n_{\text{cont}}(G, M) = \varinjlim_{U} H^n(G/U, M^U)$

where the limit is over open normal subgroups $U \triangleleft G$ .

Remark

Continuous cohomology of profinite groups is essential in étale cohomology and arithmetic geometry, providing the link between algebraic and topological methods.