Since
\(\Cor \circ \Res = [G:H]\) (
Example 3.2.24),
\(\Res(\gamma)\) generates
\(H^2(H,\Res^G_H M)\) for any
\(H\text{.}\) We start out by explicitly constructing a
\(G\)-module containing
\(M\) in which
\(\gamma\) becomes a coboundary.
Choose a 2-cocycle \(\phi\colon G^3 \to M\) representing \(\gamma\text{;}\) by the definition of a cocycle,
\begin{gather*}
\phi(g_0 g, g_1 g, g_2 g) = \phi(g_0, g_1, g_2)^g\\
\phi(g_1, g_2, g_3) - \phi(g_0, g_2, g_3) + \phi(g_0, g_1, g_3) - \phi(g_0, g_1, g_2) = 0\text{.}
\end{gather*}
Moreover, \(\phi\) is a coboundary if and only if it is of the form \(d(\rho)\text{,}\) that is, \(\phi(g_0, g_1, g_2) = \rho(g_1, g_2) - \rho(g_0, g_2) + \rho(g_0, g_1)\text{.}\) This \(\rho\) must itself be \(G\)-invariant: \(\rho(g_0, g_1)^g = \rho(g_0g, g_1g)\text{.}\) Thus \(\phi\) is a coboundary if and only if
\begin{equation*}
\phi(e, g, hg) = \rho(e,h)^g - \rho(e,hg) + \rho(e,g)\text{.}
\end{equation*}
Let \(M[\phi]\) be the direct sum of \(M\) with the free abelian group with one generator \(x_g\) for each element \(g\) of \(G - \{e\}\text{,}\) with the \(G\)-action
\begin{equation*}
x_h^g = x_{hg} - x_g + \phi(e, g, hg)\text{.}
\end{equation*}
(The symbol \(x_e\) should be interpreted as the element \(\phi(e,e,e)\) of \(M\text{.}\)) Using the cocycle property of \(\phi\text{,}\) one may verify that this is indeed a \(G\)-action; by construction, the cocycle \(\phi\) becomes zero in \(H^2(G, M[\phi])\) by setting \(\rho(e,g) = x_g\text{.}\) (Milne calls \(M[\phi]\) the splitting module of \(\phi\text{.}\)) Moreover, by the same token, for any \(H\text{,}\) the restriction of \(\phi\) to \(H\) also becomes zero in \(H^2(H, \Res^G_H M[\phi])\text{.}\)
The map \(\alpha\colon M[\phi] \to \ZZ[G]\) sending \(M\) to zero and \(x_g\) to \([g]-1\) is a homomorphism of \(G\)-modules. Actually it maps into the augmentation ideal \(I_G\text{,}\) and the sequence
\begin{equation*}
0 \to M \to M[\phi] \to I_G \to 0
\end{equation*}
is exact. Meanwhile, we also have the usual exact sequence
\begin{equation*}
0 \to I_G \to \ZZ[G] \to \ZZ \to 0\text{.}
\end{equation*}
For any subgroup \(H\) of \(G\text{,}\) we can restrict to \(H\)-modules (which we leave out of the notation for brevity), then take the long exact sequence. In the second case, since \(\ZZ[G]\) is induced (for \(G\) and hence for \(H\)), its Tate groups all vanish and we get a dimension shift:
\begin{align*}
H^1(H, I_G) &\cong H^0_T(H, \ZZ) = \ZZ/(\#H)\ZZ\\
H^2(H, I_G) &\cong H^1(H, \ZZ) = 0\text{.}
\end{align*}
In the first case, we get
\begin{gather*}
0 = H^1(H, M) \to H^1(H, M[\phi]) \to H^1(H, I_G)\\
\to H^2(H, M) \to H^2(H, M[\phi]) \to H^2(H, I_G) = 0\text{.}
\end{gather*}
Moreover, the map into \(H^2(H, M[\phi])\) vanishes by construction, so in fact \(H^2(H, M[\phi]) = 0\text{.}\) Now \(H^1(H, I_G) \to H^2(H,M)\) is a surjective map between two finite (even cyclic) groups of the same order \(\#H\text{,}\) so the map is also injective and \(H^1(H, M[\phi]) = 0\text{.}\)
We may now apply
Lemma 4.3.3 to conclude that
\(H^i_T(G, M[\phi]) = 0\) for all
\(i\text{.}\) This allows us to use the four-term exact sequence
\begin{equation*}
0 \to M \to M[\phi] \to \ZZ[G] \to \ZZ \to 0
\end{equation*}
(as in the proof of
Theorem 3.4.1) to obtain the dimension-shifting isomorphism
\(H^i_T(G, \ZZ) \cong H^{i+2}_T(G, M)\text{.}\)