Normal structure of isotropic reductive groups over rings

Abstract

The paper studies the lattice of subgroups of an isotropic reductive group G(R) over a commutative ring R, normalized by the elementary subgroup E(R). We prove the sandwich classification theorem for this lattice under the assumptions that the reductive group scheme G is defined over an arbitrary commutative ring, its isotropic rank is at least 2, and the structure constants are invertible in R. The theorem asserts that the lattice splits into a disjoint union of sublattices (sandwiches) E(R,q)<=...<=C(R,q) parametrized by the ideals q of R, where E(R,q) denotes the relative elementary subgroup and C(R,q) is the inverse image of the center under the natural homomorphism G(R) to G(R/I). The main ingredients of the proof are the "level computation" by the first author and the universal localization method developed by the second author.