On retract rationality for finite connected group schemes

TL;DR

This paper establishes the retract rationality of classifying spaces for certain finite connected group schemes over algebraically closed fields of positive characteristic, including those related to generalized Witt algebras.

Contribution

It proves retract rationality for classifying spaces of specific finite simple group schemes associated with generalized Witt algebras and extends Witt--Ree algebra concepts to general base rings.

Findings

Proved retract rationality for classifying spaces of several finite connected group schemes.

Analyzed automorphism group schemes of generalized Witt algebras.

Extended Witt--Ree algebra to general base rings.

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.