# 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 prime ideal $\mathfrak{P}$ of $K$).

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

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

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

 $\mathcal{M}=\{k\in K_{\nu}\mid\nu(k)>0\}$

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

 $k_{\nu}=\mathcal{O}_{\nu}/\mathcal{M}$

We will consider three different global Galois groups, namely

 $G_{\overline{K}/K}=\operatorname{Gal}(\overline{K}/K)$
 $G_{\overline{K_{\nu}}/K_{\nu}}=\operatorname{Gal}(\overline{K_{\nu}}/K_{\nu})$
 $G_{\overline{k_{\nu}}/k_{\nu}}=\operatorname{Gal}(\overline{k_{\nu}}/k_{\nu})$

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

 $I_{\nu}\subseteq G_{\overline{K_{\nu}}/K_{\nu}}$
###### Definition 1.

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

 $\sigma(s)=s\quad\forall\sigma\in I_{\nu},\quad\forall s\in\mathcal{S}$

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

 $I_{\nu}\cong\operatorname{Gal}(\overline{K_{\nu}}/K_{\nu}^{\operatorname{nr}})$
Title unramified action UnramifiedAction 2013-03-22 13:56:26 2013-03-22 13:56:26 alozano (2414) alozano (2414) 5 alozano (2414) Definition msc 11S15 set is unramified at a valuation InfiniteGaloisTheory DecompositionGroup Valuation