Roozbeh Hazrat

K1 of Chevalley groups are nilpotent

Abstract Let Φ be a reduced irreducible root system and R be a commutative ring. Further, let G ( Φ , R ) be a Chevalley group of type Φ over R and E ( Φ , R ) be its elementary subgroup. We prove that if the rank of Φ is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G ( Φ , R )/ E ( Φ , R ) is nilpotent by abelian. In particular, when G ( Φ , R ) is simply connected the quotient K 1 ( Φ , R )= G ( Φ , R )/ E ( Φ , R ) is nilpotent. This result was previously established by Bak for the series A 1 and by Hazrat for C 1 and D 1 . As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.

Dimension Theory and Nonstable K1 of Quadratic Modules

Employing Bak’s dimension theory, we investigate the nonstable quadratic K-group K1,2n(A, ) = G2n(A, )/E2n(A, ), n 3, where G2n(A, ) denotes the general quadratic group of rank n over a form ring (A, ) and E2n(A, ) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G2n(A, ) ⊇ G2n(A, ) ⊇ G2n(A, ) ⊇ · · · ⊇ E2n(A, ) of the general quadratic group G2n(A, ) such that G2n(A, )/G 0 2n(A, ) is Abelian, G 0 2n(A, ) ⊇ G2n(A, ) ⊇ · · · is a descending central series, and G 2n (A, ) = E2n(A, ) whenever d(A) = (Bass–Serre dimension of A) is finite. In particular K1,2n(A, ) is solvable when d(A) < ∞. Mathematics Subject Classifications (2000): 20G15, 20G35.

The yoga of commutators

In the present paper, we discuss some recent versions of localization methods for calculations in the groups of points of algebraic-like and classical-like groups. Namely, we describe relative localization, universal localization, and enhanced versions of localization-completion. Apart from the general strategic description of these methods, we state some typical technical results of conjugation calculus and commutator calculus. Also, we state several recent results obtained therewith, such as relative standard commutator formulas, bounded width of commutators with respect to the elementary generators, and nilpotent filtrations of congruence subgroups. Overall, this shows that localization methods can be much more efficient than expected. Bibliography: 74 titles.

Generalized Commutator Formulas

Let A be an algebra which is a direct limit of module finite algebras over a commutative ring R with 1. Let I, J be two-sided ideals of A, GL n (A, I) the principal congruence subgroup of level I in GL n (A), and E n (A, I) the relative elementary subgroup of level I. Using Baks localization-patching method, we prove the commutator formula which is a generalization of the standard commutator formular. This answers a problem posed by Stepanov and Vavilov.

Wedderburn's factorization theorem application to reduced K-theory

This article provides a short and elementary proof of the key theorem of reduced K-theory, namely Platonovs Congruence theorem. Our proof is based on Wedderburns factorization theorem.

Multiple commutator formulas for unitary groups

Let (A,Λ) be a formring such that A is quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. We consider the hyperbolic Bak’s unitary groups GU(2n, A, Λ), n ≥ 3. For a form ideal (I, Γ) of the form ring (A, Λ) we denote by EU(2n, I, Γ) and GU(2n, I, Γ) the relative elementary group and the principal congruence subgroup of level (I, Γ), respectively. Now, let (Ii, Γi), i = 0,...,m, be form ideals of the form ring (A, Λ). The main result of the present paper is the following multiple commutator formula: [EU(2n, I0, Γ0),GU(2n, I1, Γ1), GU(2n, I2, Γ2),..., GU(2n, Im, Γm)] =[EU(2n, I0, Γ0), EU(2n, I1, Γ1), EU(2n, I2, Γ2),..., EU(2n, Im, Γm)], which is a broad generalization of the standard commutator formulas. This result contains all previous results on commutator formulas for classicallike groups over commutative and finite-dimensional rings.

Generation of relative commutator subgroups in Chevalley groups

Let

K-Theory of Azumaya Algebras over Schemes

\Phi

Commutator width in Chevalley groups

be a reduced irreducible root system of rank

On Quillen's calculation of graded K-theory

\ge 2