Break a group into a normal subgroup and a quotient
For 1 → N → G → Q → 1 there is a first-quadrant spectral sequence with E2p,q = Hp(Q, Hq(N, A)) converging to Hp+q(G, A). It is the Grothendieck spectral sequence for taking N-invariants and then Q-invariants, and it is how essentially every non-trivial group cohomology computation is organised.
Learning objectives
- State the spectral sequence and its E2 page.
- Explain the Q-action on the cohomology of N.
- Extract the five-term sequence.
- Apply it to compute a cohomology group.
Section 01The statement
It is the Grothendieck spectral sequence for the composite A ↦ AN ↦ (AN)Q = AG. The acyclicity hypothesis holds because ℤ[G] is free over ℤ[N], so induced modules are acyclic.
Hq(N, A) carries an action of Q = G/N, because N acts trivially on its own cohomology by an inner-automorphism argument. Without that action the E2 page would not make sense — the outer cohomology is taken with respect to it.
Section 02What it gives immediately
Five-term sequence
The inflation–restriction sequence, valid with no spectral sequence machinery and sufficient for most degree-2 questions.
N with trivial cohomology
If Hq(N, A) = 0 for q > 0, the sequence collapses to the bottom row and H*(G, A) = H*(Q, AN).
Q with trivial cohomology
If Q has cohomological dimension 0 the sequence collapses to the left column, giving H*(G, A) = H*(N, A)Q.
- Choose a normal subgroup whose cohomology is known — often abelian or cyclic.
- Compute Hq(N, A) as a Q-module. Determining the action is usually the hardest step.
- Compute Hp(Q, −) of each of those, giving the E2 page.
- Determine which differentials vanish — by degree reasons, naturality, or comparison.
- Read off E∞ and solve the extension problems in each total degree.
Section 03Cautions
d2 is the transgression and can be computed in low degrees; higher differentials often require independent information about G.
Even with full collapse, the filtration of Hn(G, A) must be resolved. Ring structure and restriction maps are the usual extra input.
It combines the conjugation action of G on N with the action on the coefficients. Assuming it is trivial when it is not produces an E2 page that is simply the wrong object, and the error propagates silently through the rest of the computation.
ReferenceFrequently asked questions
Why is it a first-quadrant spectral sequence?
Because both cohomology theories vanish in negative degrees. That guarantees convergence with no additional hypothesis, which is one reason the sequence is so widely usable.
Is there a homology version?
Yes, with E²p,q = Hp(Q, Hq(N, A)) converging to Hp+q(G, A). Its five-term sequence gives Hopf's formula as a special case.
What if N is central?
The action of Q on Hq(N, A) may still be non-trivial through the coefficients, but for trivial coefficients it is trivial and the E2 page simplifies considerably. Central extensions are the standard first application.
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