I am reading the following paper on homotopy obstruction and am confused by the following notation for number fields:
On page 25, it defines
Let $K$ be a number field, $K_v$ a cmpletion of $K$.
- $\Gamma_K$ is absolute galois group of $K$.
- $I_v <\Gamma_v < \Gamma_K$ be the decomposition group and inertia
groups, and $\Gamma^{un}_v$ the unramified Galois group.
Firstly, I suppose $v$ is a choice of place here.
Question: What is the definition of $I_v, \Gamma_v$?
The definition of decomposition group and inertia group I am using are from this notes, page 5 and 16 respectively. I am confused by what, say,
the inertia group $I_v$ of $K_v$
is. If I follow the notes by Youcis, it is the kenel of a map $Gal(L/W) \rightarrow Gal(k_l/k_W)$, where $W$ is a local field, and $L$ is a Galois extension. Supposing our choice of $W$ is $K_v$, what is the choice of $L$?
Best Answer
$\mathfrak{p}$ is a prime ideal of $O_K$ corresponding to $v$, $O_{K_v}=\varprojlim O_K/\mathfrak{p}^n$,$K_v=O_{K_v}[\pi_{K_v}^{-1}]$
$L= \overline{K_v}$ (into which $\overline{K}$ is dense whatever embedding you choose), $k_l=O_{\overline{K_v}}/\mathfrak{m}_{\overline{K_v}}$ (an algebraic closure of $k_w$),$k_w=O_{K_v}/(\pi_{K_v})$. Then $I_v=\ker(Gal(L/W) \rightarrow Gal(k_l/k_W))$ and $\Gamma_v = Gal(L/W) =D_\mathfrak{P}$ where $\mathfrak{P}$ is a prime ideal of $O_{\overline{K}}$ above $v$ and $D_\mathfrak{P}$ is the subgroup of $Gal(\overline{K}/K)$ which is continuous for (the valuation corresponding to) $\mathfrak{P}$.