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

Stasinski, Alexander (2010) Reductive group schemes, the Greenberg functor, and associated algebraic groups. Preprint, 1-14.

Abstract

Let A be an Artinian local ring with algebraically closed residue field k, and let G be an affine smooth group scheme over A. The Greenberg functor F associates to G a linear algebraic group G := (FG)(k) over k, such that G = G(A). We prove that if G is a reductive group scheme over A, and T is a maximal torus of 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 G is reductive and P is a parabolic subgroup of G, then P is a self-normalising subgroup of G, and if B and B0 are two Borel subgroups of G, then the corresponding subgroups B and B0 are conjugate in G

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