Lemma 58.13.6. Let $A$ be a discrete valuation ring with fraction field $K$. Let $K^{sep}$ be a separable closure of $K$. Let $A^{sep}$ be the integral closure of $A$ in $K^{sep}$. Let $\mathfrak m^{sep}$ be a maximal ideal of $A^{sep}$. Let $\mathfrak m = \mathfrak m^{sep} \cap A$, let $\kappa = A/\mathfrak m$, and let $\overline{\kappa } = A^{sep}/\mathfrak m^{sep}$. Then $\overline{\kappa }$ is an algebraic closure of $\kappa $. Let $G = \text{Gal}(K^{sep}/K)$, $D = \{ \sigma \in G \mid \sigma (\mathfrak m^{sep}) = \mathfrak m^{sep}\} $, and $I = \{ \sigma \in D \mid \sigma \bmod \mathfrak m^{sep} = \text{id}_{\kappa (\mathfrak m^{sep})}\} $. The decomposition group $D$ fits into a canonical exact sequence

where $\kappa ^{sep} \subset \overline{\kappa }$ is the separable closure of $\kappa $. The inertia group $I$ fits into a canonical exact sequence

such that

$P$ is a normal subgroup of $D$,

$P$ is a pro-$p$-group if the characteristic of $\kappa _ A$ is $p > 1$ and $P = \{ 1\} $ if the characteristic of $\kappa _ A$ is zero,

there exists a canonical surjective map

\[ \theta _{can} : I \to \mathop{\mathrm{lim}}\nolimits _{n\text{ prime to }p} \mu _ n(\kappa ^{sep}) \]whose kernel is $P$, which satisfies $\theta _{can}(\tau \sigma \tau ^{-1}) = \tau (\theta _{can}(\sigma ))$ for $\tau \in D$, $\sigma \in I$, and which induces an isomorphism $I_ t \to \mathop{\mathrm{lim}}\nolimits _{n\text{ prime to }p} \mu _ n(\kappa ^{sep})$.

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