Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On retract rationality for finite connected group schemes

Shusuke Otabe

TL;DR

This work investigates retract rationality of classifying spaces $BG$ for finite connected group schemes over algebraically closed fields of positive characteristic. It develops a framework based on triangulating automorphism group schemes of generalized Witt algebras, and extends Witt–Ree algebra theory to general base rings to handle twisted forms and torsor lifting. The authors prove retract rationality for several important families, notably Frobenius kernels of special groups and the finite simple group schemes $\Gamma(m;\bm{n})$ in cases $\bm{n}=\bm{1}$ or $m=1$, by reducing to surjectivity of certain $H^1_{\mathrm{fppf}}$-maps and leveraging explicit decompositions. The results advance Noether-type rationality questions in positive characteristic for finite connected group schemes and provide a structural toolkit (triangulations, WR-algebras, and lifting criteria) for further progress.

Abstract

In the present paper, we prove the retract rationality of the classifying spaces $BG$ for several types of finite connected group schemes $G$ over algebraically closed fields of positive characteristic $p>0$. In particular, we prove the retract rationality for the finite simple group schemes $G$ associated with the generalized Witt algebras in specific cases. To this end, we study the automorphism group schemes of the generalized Witt algebras and establish triangulations for them. Moreover, we extend the notion of Witt--Ree algebra to general base rings and discuss their properties.

On retract rationality for finite connected group schemes

TL;DR

This work investigates retract rationality of classifying spaces for finite connected group schemes over algebraically closed fields of positive characteristic. It develops a framework based on triangulating automorphism group schemes of generalized Witt algebras, and extends Witt–Ree algebra theory to general base rings to handle twisted forms and torsor lifting. The authors prove retract rationality for several important families, notably Frobenius kernels of special groups and the finite simple group schemes in cases or , by reducing to surjectivity of certain -maps and leveraging explicit decompositions. The results advance Noether-type rationality questions in positive characteristic for finite connected group schemes and provide a structural toolkit (triangulations, WR-algebras, and lifting criteria) for further progress.

Abstract

In the present paper, we prove the retract rationality of the classifying spaces for several types of finite connected group schemes over algebraically closed fields of positive characteristic . In particular, we prove the retract rationality for the finite simple group schemes associated with the generalized Witt algebras in specific cases. To this end, we study the automorphism group schemes of the generalized Witt algebras and establish triangulations for them. Moreover, we extend the notion of Witt--Ree algebra to general base rings and discuss their properties.
Paper Structure (10 sections, 25 theorems, 86 equations)

This paper contains 10 sections, 25 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Trigonalizable group schemes
  3. Preliminaries
  4. Frobenius kernels of special algebraic groups
  5. Trigonalizable group schemes
  6. Simple group schemes associated with the generalized Witt algebras
  7. The automorphisms of the generalized Witt algebras
  8. Triangulation of the automorphism group schemes
  9. The Witt--Ree algebras over general base rings
  10. Retract rationality for the generalized Witt algebras

Key Result

Theorem 1.2

(cf. Theorem thm: ret rat trig) Let $G$ be a finite $k$-group scheme. If $G$ is trigonalizable, the associated classifying space $BG$ is retract rational over $k$.

Theorems & Definitions (66)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • ...and 56 more