Definition 15.112.3. Let $A$ be a discrete valuation ring with fraction field $K$. Let $L/K$ be a finite Galois extension with Galois group $G$. Let $B$ be the integral closure of $A$ in $L$. Let $\mathfrak m \subset B$ be a maximal ideal.

1. The decomposition group of $\mathfrak m$ is the subgroup $D = \{ \sigma \in G \mid \sigma (\mathfrak m) = \mathfrak m\}$.

2. The inertia group of $\mathfrak m$ is the kernel $I$ of the map $D \to \text{Aut}(\kappa (\mathfrak m)/\kappa _ A)$.

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