2010/03/18 by Alexander Stasinski, Stasinski, Alexander
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Alkaloids: synthesis and pharmacology #FOS: Mathematics #math.AG
paper · pdf · doi:10.48550/arxiv.1003.3598
Minor corrections; see the errata notes
openalex publication_date 2010/03/18 · arxiv created 2014/03/07 · arxiv updated 2014/03/10 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
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\congG(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 B' are two Borel subgroups of G, then the corresponding subgroups B and B' are conjugate in G.