Reductive group schemes, the Greenberg functor, and associated algebraic groups

TL;DR

This paper investigates the relationship between reductive group schemes over Artinian local rings and their associated algebraic groups via the Greenberg functor, establishing correspondences for maximal tori, Borel, and parabolic subgroups.

Contribution

It proves that maximal tori correspond to Cartan subgroups and Borel subgroups are conjugate in the associated algebraic group, extending classical results to this setting.

Findings

Maximal tori in reductive group schemes correspond to Cartan subgroups in the algebraic group.

Borel subgroups are conjugate within the associated algebraic group.

Parabolic subgroups are self-normalising in the algebraic group.

Abstract

Let $A$ be an Artinian local ring with algebraically closed residue field $k$, and let $\mathbf{G}$ be an affine smooth group scheme over $A$. The Greenberg functor $\mathcal{F}$ associates to $\mathbf{G}$ a linear algebraic group $G:=(\mathcal{F}\mathbf{G})(k)$ over $k$, such that $G\cong\mathbf{G}(A)$. We prove that if $\mathbf{G}$ is a reductive group scheme over $A$, and $\mathbf{T}$ is a maximal torus of $\mathbf{G}$, then $T$ is a Cartan subgroup of $G$, and every Cartan subgroup of $G$ is obtained uniquely in this way. The proof is based on establishing a Nullstellensatz analogue for smooth affine schemes with reduced fibre over $A$, and that the Greenberg functor preserves certain normaliser group schemes over $A$. Moreover, we prove that if $\mathbf{G}$ is reductive and $\mathbf{P}$ is a parabolic subgroup of $\mathbf{G}$, then $P$ is a self-normalising subgroup of $G$, and if $\mathbf{B}$ and $\mathbf{B}'$ are two Borel subgroups of $\mathbf{G}$, then the corresponding subgroups $B$ and $B'$ are conjugate in $G$.