I am reading the notes of J.S. Milne on the Class Field Theory. http://www.jmilne.org/math/CourseNotes/cft.html
And I have difficulty understanding the proof of the Tate's Theorem. On the page 66, it reads $$ 0 \to H^1(H,C(\phi)) \to H^1(H,I_G) \to H^2(H,C) \stackrel{0}{\to} H^2(H,C(\phi)) \to 0, $$ and I don't understand why $H^2(H,C) \to H^2(H,C(\phi))$ is a zero map. It also says that $\operatorname{Res}(\gamma)$, which is a generator of the cyclic group $H^2(H,C)$ is mapped to zero, but I have no idea why this is the case.
Thanks for advance.
Res($\gamma$) maps to the image of $\gamma$ in $H^2(H,C(\phi))$, which is shown to be zero at the top of the same page (p.82 in the current version of the notes). In fact $C(\phi)$ is exactly defined so that $\gamma$ maps to zero in $H^2(H,C(\phi))$, which is why it is called the splitting module.