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Free Proalgebraic Groups

Michael Wibmer

Abstract

Replacing finite groups by linear algebraic groups, we study an algebraic-geometric counterpart of the theory of free profinite groups. In particular, we introduce free proalgebraic groups and characterize them in terms of embedding problems. The main motivation for this endeavor is a differential analog of a conjecture of Shafarevic.

Free Proalgebraic Groups

Abstract

Replacing finite groups by linear algebraic groups, we study an algebraic-geometric counterpart of the theory of free profinite groups. In particular, we introduce free proalgebraic groups and characterize them in terms of embedding problems. The main motivation for this endeavor is a differential analog of a conjecture of Shafarevic.

Paper Structure

This paper contains 14 sections, 46 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Free pro-$\mathcal{C}$-groups
  3. Preliminaries and notation
  4. Formations of algebraic groups
  5. Coalgebraic subgroups and convergence to one
  6. Generating proalgebraic groups
  7. Existence of free pro-$\mathcal{C}$-groups
  8. Saturated pro-$\mathcal{C}$-groups
  9. The rank of a proalgebraic group
  10. The dimension of a proalgebraic group
  11. Embedding problems
  12. Existence of saturated pro-$\mathcal{C}$-groups
  13. Uniqueness of saturated pro-$\mathcal{C}$-groups
  14. Free pro-$\mathcal{C}$-groups and saturation

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field. Then the absolute Galois group of $K=k(x)$, the field of rational functions over $k$, is the free profinite group on a set of cardinality $|K|$.

Theorems & Definitions (120)

  • Theorem 1.1: Douady, Pop, Harbater
  • Conjecture : Matzat
  • Theorem 1.2: BachmayrHarbaterHartmannWibmer:FreeDifferentialGaloisGroups
  • Theorem 1.3: Corollary \ref{['cor: Iwasawa for algebaic groups']}
  • Theorem 1.4: Theorem \ref{['theo: free=saturated']}
  • Remark 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 110 more