ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: Banaszak, Grzegorz Wild kernel for higher K-groups and arithmetic of square rings In a joint work with W. Gajda, P. Krason and P. Zelewski we define in a natural way a subgroup W K_{n}(F) of K_{n}(F), called “wild kernel”. This subgroup is always finite and we conjecture that:

W K_{n}(F) = divK_{n}(F)

up to 2-torsion, for all non-negative n and all number fields F. divK_{n}(F) denotes the subgroup of divisible elements in K_{n}(F). The case n = 2 was considered by Tate and Bass many years ago. It turns out that our conjecture is equivalent to Quillen-Lichtenbaum conjecture for F = Q and n > 1. The equivalence for other number fields is proven under some condition on K - groups of local fields. We use a description of higher K-groups as SK_{1}(F_{n}) where F_{n} is the square ring, to investigate further the wild kernel W K_{n}(F).