On projective modules for Frobenius kernels and finite Chevalley groups

Details On projective modules for Frobenius kernels and finite Chevalley groups On projective modules for Frobenius kernels and finite Chevalley groups

2012-04-03

arxiv.org/abs/1204.0729v1

Mathematics

Representation Theory

7 pages

Scientific paper

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 $G$.

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