On the local behaviour of specializations of function field extensions

TL;DR

This paper studies how primes in a base field behave locally when specializing Galois extensions of function fields, extending known results on inertia groups and exploring implications for crossed products and parametric sets.

Contribution

It generalizes Beckmann's results on inertia groups to decomposition groups in specializations of Galois extensions over function fields.

Findings

Extended understanding of decomposition groups in specializations

Applied results to crossed products and Hilbert--Grunwald property

Provided insights into finite parametric sets

Abstract

Given a field $k$ of characteristic zero and an indeterminate $T$ over $k$, we investigate the local behaviour at primes of $k$ of finite Galois extensions of $k$ arising as specializations of finite Galois extensions $E/k(T)$ (with $E/k$ regular) at points $t_0 \in \mathbb{P}^1(k)$. We provide a general result about decomposition groups at primes of $k$ in specializations, extending a fundamental result of Beckmann concerning inertia groups. We then apply our result to study crossed products, the Hilbert--Grunwald property, and finite parametric sets.