Date & Time: November 09, 2011, 14:30-15:30.
Venue:Hall No. 216 (Second Floor)
Title: Chevalley groups over rings: universal localisation
Speaker: Prof. Alexei Stepanov, University Of Illinois, At Urbana-Champaign
Abstract: CHEVALLEY GROUPS OVER RINGS: UNIVERSAL LOCALIZATION ALEXEI STEPANOV
Beginning from works of Suslin and Quillen on Serre's conjecture, localization methods proved their importance in the theory of algebraic groups over rings. I shall talk on a new version of a localization mehtod. The idea is to use localization in a \universal" ring U, e. g. the ane algebra of a groups G, get a result in G(U), and than project it to G(R) for an arbitrary ring R. Clearly, the results obtained in this way does not depend on R. For example, we obtain the following theorem.
Theorem: Let G be a Chevalley{Demazure group scheme with a root system of rank 2, and let E be its elemetary subgroup subfunctor. Then there exists a constant L = L(G) such that for any ring R and any elements a 2 G(R) and b 2 E(R) the commutator [a; b] is a product of at most L elementary root unipotent elements.
Let me explain the importance of this result. Width of a group H with respect to a generating set S is the smallest integer L (or in nity) such that any element of H decomposes in a product of at most L generators. The width of the linear elementary group En(R) with repect to elementary generators or the set of all commutators was studied by Carter, Keller, Dennis, Vaserstein, van der Kallen and others. It is concerned with computing of the Kazhdan constatant. For example, the width of E(R) is nite if R is semilocal (by Gauss decomposition) or R = Z (Carter, Keller), but is in nite for R = C[x] (van der Kallen). And answer is unknown already for R = F[x] for a nite eld F.
Van der Kallen noticed that the group En(R)1=En(R1) is an obstruction for the nitness of width of En(R), were in nite power means the direct product of countably many copies of a ring or a group. The theorem above is equivalent to the fact that this group is central in K1(R1), so one can study it using homological algebra.
During the proof we obtain the standard commutator formulas with an estimate of width of con- jugates with elementary root unipotents and commutators. The proof can be applied to establish the nilpotent structure of K1.
The proof bases on Gauss decomposition, elementary calculations and easy splitting arguments. This gives a hope to extend main structure theorems for Chevalley groups over rings (commutator formulas, normal structure, niloptent structure of K1, width of commutators) to nonsplit isotropic reductive groups and generalized congruence subgroups, e. g. generalized unitary groups of A. Bak.