A note on the Quillen-Lichtenbaum conjecture and the arithmetic of square rings, by Grzegorz Banaszak, Wojciech Gajda, Piotr Krason, and Piotr Zelewski

We define a wild kernel for K-groups of number fields for all indices. This wild kernel is always finite. We conjecture that the wild kernel in K_n(F) should be equal to divisible elements in K_n(F), up to 2-torsion. We can verify this conjecture for n = 0, 1, 2, 3 and for any number field F, and also for n = 4 and F = Q, hence in the cases where K-groups are better understood. We give a condition when our conjecture and the conjecture of Quillen-Lichtenbaum are equivalent. One of our results shows that these conjectures are equivalent - unconditionally - for F = Q and n > 1. In the last two sections of our paper we work with SK_1 description of K-theory. We reformulate our conjecture in terms of linear algebra over square rings, and investigate further the wild kernel.

Grzegorz Banaszak <banaszak@math.amu.edu.pl>
Wojciech Gajda <gajda@math.amu.edu.pl>
Piotr Krason <krason@uoo.univ.szczecin.pl>
Piotr Zelewski <piotr@icarus.math.mcmaster.ca>