unramified action

Let $K$ be a number field and let $\nu$ be a discrete valuation on$K$ (this might be, for example, the valuation attached to a primeideal $𝔓$ of $K$).

Let $K_{\nu }$ be the completion of $K$ at $\nu$, and let${𝒪}_{\nu }$ be the ring of integers of $K_{\nu }$, i.e.

$${𝒪}_{\nu }=\{k\in K_{\nu }\mid \nu (k)\ge 0\}$$

The maximal ideal of ${𝒪}_{\nu }$ will be denoted by

$$ℳ=\{k\in K_{\nu }\mid \nu (k)>0\}$$

and we denote by $k_{\nu }$ the residue field of $K_{\nu }$, whichis

$$k_{\nu }={𝒪}_{\nu }/ℳ$$

We will consider three different global Galois groups, namely

$$G_{\overline{K}/K}=\mathrm{Gal}(\overline{K}/K)$$

$$G_{\overline{K_{\nu }}/K_{\nu }}=\mathrm{Gal}(\overline{K_{\nu }}/K_{\nu })$$

$$G_{\overline{k_{\nu }}/k_{\nu }}=\mathrm{Gal}(\overline{k_{\nu }}/k_{\nu })$$

where $\overline{K},\overline{K_{\nu }},\overline{k_{\nu }}$ are algebraic closures of the corresponding field. We alsodefine notation for the inertia group of$G_{\overline{K_{\nu }}/K_{\nu }}$

$$I_{\nu }\subseteq G_{\overline{K_{\nu }}/K_{\nu }}$$

Definition 1.

Let $\mathrm{S}$ be a set and suppose there is a group action of$G\mathit{}a\mathit{}l\mathit{}\mathrm{(}\overline{K_{\nu }}\mathrm{/}{K}_{\nu }\mathrm{)}$ on $\mathrm{S}$. We say that$\mathrm{S}$ is unramified at $\nu$, or the action of$G_{\overline{K_{\nu }}\mathrm{/}{K}_{\nu }}$ on $\mathrm{S}$ is unramified at$\nu$, if the action of $I_{\nu }$ on $\mathrm{S}$ is trivial,i.e.

$$\sigma (s)=s\mathit{\hspace{1em}}\forall \sigma \in I_{\nu },\forall s\in 𝒮$$

Remark: By Galois theory we know that,$K_{\nu }^{\mathrm{nr}}$, the fixed field of $I_{\nu }$, theinertia subgroup, is the maximal unramified extension of$K_{\nu }$, so

$$I_{\nu }\cong \mathrm{Gal}(\overline{K_{\nu }}/K_{\nu }^{\mathrm{nr}})$$