There is a statement in SGA 7-1 Exposé 1 (P. Deligne, Résumé des premiers exposés de A. Grothendieck, pdf of SGA7-1), (0.3.1):

$S$ is a Henselian trait (i.e. the spectrum of a henselian discrete valuation ring), $\eta$, $s$, $V$, $\bar{V}$, $\bar{\eta}$, $\bar{s}$ as usual, and $1\to I\to Gal(\bar{\eta}/\eta) \to Gal(\bar{s}/s) \to 1$. $p$ is a characteristic of the residue field of $s$, $P$ be the pro-$p$ part of $I$, then the tame ramification group $I/P$ is canonically isomorphic to $\hat{\mathbb{Z}}(1)(k(\bar{s}))$.

Then the action of $Gal(\bar{s}/s)$ on $I/P$ can be identified with the natural action of $Gal(\bar{s}/s)$ on $\hat{\mathbb{Z}}(1)(k(\bar{s}))$.

It refers to Serre's Local Fields for proofs, but I cannot come up with a proof of this statement out of the this reference. Can somebody elaborate on this, or refer to a relevant section of the book?


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