On projective modules for Frobenius kernels and finite Chevalley groups

TL;DR

This paper proves that projectivity of rational modules over Frobenius kernels implies projectivity over finite Chevalley groups for semisimple algebraic groups, extending prior results and clarifying limitations for unipotent radicals.

Contribution

It establishes a link between projectivity over Frobenius kernels and finite Chevalley groups for semisimple groups, extending previous theorems and identifying cases where the property fails.

Findings

Projectivity over Frobenius kernels implies projectivity over finite Chevalley groups for semisimple groups.

The result does not hold when replacing the group with the unipotent radical of a Borel subgroup.

The theorem of Lin and Nakano is salvaged and extended to broader contexts.

Abstract

Let $G$ be a simply-connected semisimple algebraic group scheme over an algebraically closed field of characteristic $p > 0$. Let $r \geq 1$ and set $q = p^r$. We show that if a rational $G$-module $M$ is projective over the $r$-th Frobenius kernel $G_r$ of $G$, then it is also projective when considered as a module for the finite subgroup $\Gfq$ of $\Fq$-rational points in $G$. This salvages a theorem of Lin and Nakano (\emph{Bull.\ London Math.\ Soc.} 39 (2007) 1019--1028). We also show that the corresponding statement need not hold when the group $G$ is replaced by the unipotent radical $U$ of a Borel subgroup of $G$.