Stasinski, Alexander (2012) 'Reductive group schemes, the Greenberg functor, and associated algebraic groups.', Journal of pure and applied algebra., 216 (5). pp. 1092-1101.
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.
|Full text:||(AM) Accepted Manuscript|
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|Publisher Web site:||http://dx.doi.org/10.1016/j.jpaa.2011.10.027|
|Publisher statement:||This is the author’s version of a work that was accepted for publication in Journal of Pure and Applied Algebra. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Alexander Stasinski, Reductive group schemes, the Greenberg functor, and associated algebraic groups, Journal of Pure and Applied Algebra, Volume 216, Issue 5, May 2012, Pages 1092-1101, http://dx.doi.org/10.1016/j.jpaa.2011.10.027.|
|Date accepted:||No date available|
|Date deposited:||06 May 2014|
|Date of first online publication:||May 2012|
|Date first made open access:||No date available|
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