The <short exact sequence> of $G$-modules induces the <long exact sequence in group cohomology>
$$
0\to H^0(G,M_1)\to H^0(G,M_2)\to H^0(G,M_3)
\xrightarrow{\delta}H^1(G,M_1)\to H^1(G,M_2)\to\cdots.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 151 / 4 / b / i / Solution

A <short exact sequence> $0\to M_1\to M_2\to M_3\to0$ of $G$-modules induces a natural long exact sequence
$$
0\to H^0(G,M_1)\to H^0(G,M_2)\to H^0(G,M_3)
\to H^1(G,M_1)\to\cdots.
$$
It follows by applying the cochain functor and the long exact cohomology sequence of a short exact sequence of cochain complexes.

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 Incoming links: Long exact sequence in group cohomology