Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 4 / a / ii

Table of contents
- Solution ii

Solution

A crossed homomorphism is a map

ϕ : G \to M

satisfying

ϕ (g h) = ϕ (g) + g ϕ (h) .

For a one-cochain, the formula in part i gives

(d ϕ) (g, h) = g ϕ (h) - ϕ (g h) + ϕ (g),

so the crossed homomorphisms are exactly the one-cocycles. A zero-cochain

m \in M

has coboundary

g \mapsto g m - m

, the principal crossed homomorphism associated with

m

. Therefore

H^{1} (G, M) = \frac{{ crossed homomorphisms G \to M }}{{ g \mapsto g m - m : m \in M }} .

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