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Versal Family of Reductive Groups

Shahar Dagan

TL;DR

The paper addresses constructing a versal, finite-type family parameterizing all connected reductive groups of rank at most $n$. It couples root datum data, finite étale Galois data, and Galois cohomology into a base scheme and builds a reductive group scheme over it, then extends from quasi-split forms to all forms via 1-cocycles. The main contributions are the existence of a finite-type base $\mathcal{S}_n$ with a smooth $\mathcal{S}_n$-group $\mathcal{R}_n$ whose fibers realize every reductive group of rank $\le n$, and a parallel versal family for quasi-split reductive groups. These results provide a uniform framework for studying invariants of reductive groups with bounded rank and extend prior finite-field work to general base fields.

Abstract

In this paper, we construct a family of reductive groups, including all reductive groups up to a given rank. We also construct a similar versal family of quasi-split reductive groups. This result generalizes a former result of N.Avni and R.Aizenbud and provides a framework for systematically studying invariants of reductive groups with bounded rank.

Versal Family of Reductive Groups

TL;DR

The paper addresses constructing a versal, finite-type family parameterizing all connected reductive groups of rank at most . It couples root datum data, finite étale Galois data, and Galois cohomology into a base scheme and builds a reductive group scheme over it, then extends from quasi-split forms to all forms via 1-cocycles. The main contributions are the existence of a finite-type base with a smooth -group whose fibers realize every reductive group of rank , and a parallel versal family for quasi-split reductive groups. These results provide a uniform framework for studying invariants of reductive groups with bounded rank and extend prior finite-field work to general base fields.

Abstract

In this paper, we construct a family of reductive groups, including all reductive groups up to a given rank. We also construct a similar versal family of quasi-split reductive groups. This result generalizes a former result of N.Avni and R.Aizenbud and provides a framework for systematically studying invariants of reductive groups with bounded rank.

Paper Structure

This paper contains 12 sections, 19 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Structure of the proof
  3. Structure of the paper
  4. Acknowledgement
  5. Preliminaries
  6. Scheme theory constructions
  7. Forms of reductive groups
  8. Reductive group schemes
  9. Proof of main results
  10. Family of finite étale algebras
  11. Versal family of quasi-split reductive groups
  12. Versal family of reductive groups

Key Result

Theorem 1

For any integer $n>0$, there exists a scheme $\mathcal{S} _n$ of finite type and a smooth $\mathcal{S}_n$-group scheme of finite type $\mathcal{R}_n\tilde{o} \mathcal{S}_n$ such that:

Theorems & Definitions (63)

  • Theorem 1: see Theorem \ref{['thm: versal family reductive']} below
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • ...and 53 more