Wild kernels and divisibility in K-groups of global fields

Authors:

• Grzegorz Marian Banaszak

Abstract

Text: In 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 ^lim1 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). 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. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=pQXdg8o4sIs. © 2013 Elsevier Inc.