Wild kernels and divisibility in K-groups of global fields

TextIn this paper we study divisibility and wild kernels in algebraic K-theory of global fields F. We extend the notion of the wild kernel to all K-groups of global fields and prove that the Quillen–Lichtenbaum conjecture for F is equivalent to the equality of wild kernels with the corresponding groups of divisible elements in K-groups of F  . We show that there exist generalized Moore exact sequences for even K-groups of global fields. Without appealing to the Quillen–Lichtenbaum conjecture we show that the group of divisible elements is isomorphic to the corresponding group of étale divisible elements and we apply this result for the proof of the lim1lim1 analogue of the Quillen–Lichtenbaum conjecture. We also apply this isomorphism to investigate: the imbedding obstructions in homology of GL  , the splitting obstructions for the Quillen localization sequence, the order of the group of divisible elements via special values of ζF(s)ζF(s). Using the motivic cohomology results due to Bloch, Friedlander, Levine, Lichtenbaum, Morel, Rost, Suslin, Voevodsky and Weibel, which established the Quillen–Lichtenbaum conjecture, we conclude that wild kernels are equal to the corresponding groups of divisible elements.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=pQXdg8o4sIs.