Lyndon–Hochschild–Serre spectral sequence

Remarks

The spectral sequence is the Grothendieck spectral sequence of the composite of the functors $M \mapsto M^N$ and $A \mapsto A^{G/N}$, whose composite is $M \mapsto M^G$. It applies because $M \mapsto M^N$ sends injective $G$-modules to $G/N$-modules that are acyclic for $A \mapsto A^{G/N}$. There is a homological version $E^2_{p,q} = H_p(G/N, H_q(N, M)) \Rightarrow H_{p+q}(G, M)$ for group homology.

The terms of low degree give the inflation-restriction exact sequence $$0 \to H^1(G/N, M^N) \to H^1(G, M) \to H^1(N, M)^{G/N} \to H^2(G/N, M^N) \to H^2(G, M).$$

Example: suppose that $H^q(N, M) = 0$ for all $q \ge 1$, as happens when $N$ is finite and multiplication by $\abs{N}$ is bijective on $M$. Then the $E_2$ page is concentrated in the row $q = 0$, the sequence collapses, and $H^p(G, M) \cong H^p(G/N, M^N)$ for all $p$.

Going deeper

For a student

The five-term exact sequence of the homological version already gives Hopf's formula for $H_2$. For a polycyclic group, one can apply the spectral sequence step by step along a normal series with cyclic factors. Its topological counterpart is the Serre spectral sequence of the fibration $BN \to BG \to B(G/N)$ of classifying spaces.