The Stacks project

Remark 15.112.11. In order to use the inertia character $\theta : I \to \mu _ e(\kappa (\mathfrak m))$ for infinite Galois extensions, it is convenient to scale it. Let $A, K, L, B, \mathfrak m, G, P, I, D, e, \theta $ be as in Lemma 15.112.5 and Definition 15.112.6. Then $e = q |I_ t|$ with $q$ is a power of the characteristic $p$ of $\kappa (\mathfrak m)$ if positive or $1$ if zero. Note that $\mu _ e(\kappa (\mathfrak m)) = \mu _{|I_ t|}(\kappa (\mathfrak m))$ because the characteristic of $\kappa (\mathfrak m)$ is $p$. Consider the map

This map induces an isomorphism $\theta _{can} : I_ t \to \mu _{|I_ t|}(\kappa (\mathfrak m))$. We have $\theta _{can}(\tau \sigma \tau ^{-1}) = \tau (\theta _{can}(\sigma ))$ for $\tau \in D$ and $\sigma \in I$ by Lemma 15.112.7. Finally, if $M/L$ is an extension such that $M/K$ is Galois and $\mathfrak m'$ is a prime of the integral closure of $A$ in $M$ lying over $\mathfrak m$, then we get the commutative diagram

by Lemma 15.112.10.