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On p.64 of the book Cyclotomic Fields and Zeta Values by J. Coates and R. Sujatha: They seemed to have used the argument as follows: Let $G$ be a pro-$p$ group. If $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is an exact sequence of $G$-modules, then taking $G$-invariant, we have an exact sequence $C^G\rightarrow A_G\rightarrow B_G$.

I don't know why, could anybody help me? Thanks.

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Great book. That's not always true in the generality you stated it, and I don't have the book on me so I don't know quite what assertion you're quoting, but if we assume everything is a $\mathbb{Z}_p[[G]]$-module, then the exactness of the sequence

$$ 0 \to A^G \to B^G \to C^G \to A_G \to B_G \to C_G \to 0 $$

follows from applying the snake lemma to the map $\gamma - 1$ from the short exact sequence $0 \to A \to B \to C \to 0$ to itself.

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