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

Abstract

Let AA be an Artinian local ring with algebraically closed residue field kk, and let View the MathML sourceG be an affine smooth group scheme over AA. The Greenberg functor FF associates to View the MathML sourceG a linear algebraic group View the MathML sourceG≔(FG)(k) over kk, such that View the MathML sourceG≅G(A). We prove that if View the MathML sourceG is a reductive group scheme over AA, and View the MathML sourceT is a maximal torus of View the MathML sourceG, then TT is a Cartan subgroup of GG, and every Cartan subgroup of GG is obtained uniquely in this way. Moreover, we prove that if View the MathML sourceG is reductive and View the MathML sourceP is a parabolic subgroup of View the MathML sourceG, then PP is a self-normalising subgroup of GG, and if View the MathML sourceB and View the MathML sourceB′ are two Borel subgroups of View the MathML sourceG, then the corresponding subgroups BB and B′B′ are conjugate in GG.