Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local-global principles for the existence of Levi factors

David Harbater, Julia Hartmann, George McNinch

Abstract

We discuss local-global principles for the existence of Levi factors (i.e., complements to the unipotent radical) for linear algebraic groups over one-variable function fields. We give examples of disconnected groups that fail the local-global principle, and prove a strong local-global principle in the presence of Levi descent.

Local-global principles for the existence of Levi factors

Abstract

We discuss local-global principles for the existence of Levi factors (i.e., complements to the unipotent radical) for linear algebraic groups over one-variable function fields. We give examples of disconnected groups that fail the local-global principle, and prove a strong local-global principle in the presence of Levi descent.

Paper Structure

This paper contains 4 sections, 5 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A counterexample to the local-global principle
  4. A strong local-global principle via Levi descent

Key Result

Proposition 3.1

With the above notation, let $P(Z)=Z^p-Z-f\in F[Z]$, and let $L/F$ be a finite extension of fields. Then the base change $(G_f)_L$ of $G_f$ to $L$ has a Levi factor if and only if $P$ has a root in $L$.

Theorems & Definitions (12)

  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Remark 3.6
  • Theorem 4.1
  • ...and 2 more