Remark 15.112.11 (0BU8)—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

$\theta _{can} = q\theta : I \longrightarrow \mu _{|I_ t|}(\kappa (\mathfrak m))$

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

$\xymatrix{ I' \ar[r]_-{\theta '_{can}} \ar[d] & \mu _{|I'_ t|}(\kappa (\mathfrak m')) \ar[d]^{(-)^{|I'_ t|/|I_ t|}} \\ I \ar[r]^-{\theta _{can}} & \mu _{|I_ t|}(\kappa (\mathfrak m)) }$

by Lemma 15.112.10.

The Stacks project

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