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

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

2010-03-18

arxiv.org/abs/1003.3598v1

Mathematics

Algebraic Geometry

14 pages

Scientific paper

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 $G$.

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