Lagmental Vicfred

The Krull Topology Uses Finite Fixed Fields as Neighbourhoods by Vicfred

tags: mathematics, infinite Galois theory, local fields, ramification

Stabilisers of finite subextensions form a neighbourhood basis of the identity. A small computation will anchor the general statement before the abstraction takes over.

Objects and notation

Infinite Galois groups carry the Krull topology and become profinite groups. For a discretely valued field \(K\), completions and residue fields add a second layer of arithmetic to extensions \(L/K\).

$$ U_E=\operatorname{Gal}(L/E) $$

The typography mirrors the proof: first declare \(\mathsf D\), then state \(\mathsf C\). The symbol \(\Longrightarrow\) below is a logical dependency, not extra mathematical structure.

$$ \{U_E:E/K\text{ finite}\}\ \text{is a basis at }1 $$

Push the symbols

The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.

$$ \overline H=\bigcap_{\substack{E/K\ \mathrm{finite}\\H\subseteq U_EH}}U_EH,\qquad L^{\overline H}=L^H $$

Structural reading

The invariant statement is the one that does not depend on a convenient choice of coordinates, representatives, basis, or enumeration.

$$ \begin{aligned} \mathsf{D}\;&:\quad U_E=\operatorname{Gal}(L/E),\\[5pt] \mathsf{C}\;&:\quad \{U_E:E/K\text{ finite}\}\ \text{is a basis at }1. \end{aligned} $$

A hypothesis worth keeping

Subgroups in infinite Galois theory correspond to intermediate fields only after taking closure. Ramification filtrations also depend on the chosen valuation.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \{U_E:E/K\text{ finite}\}\ \text{is a basis at }1 \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Sat 30 May 2020. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.