Wild Kernels and divisibility in K-groups of global fields

TL;DR

This paper explores the relationship between wild kernels and divisible elements in algebraic K-theory of global fields, extending known conjectures and establishing new isomorphisms and exact sequences.

Contribution

It generalizes the notion of wild kernels to all K-groups of global fields and proves their equality with divisible elements under certain conjectures, providing new tools for K-theory analysis.

Findings

Wild kernels equal divisible elements under Quillen-Lichtenbaum conjecture.

Existence of generalized Moore exact sequences for even K-groups.

Isomorphism between divisible elements and étale divisible elements.

Abstract

In this paper we study the divisibility and the 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 Quillen-Lichtenbaum conjecture for $F$ is equivalent to the equality of wild kernels with 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 \' etale divisible elements and we apply this result for the proof of the $lim^1$ analogue of 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 $\zeta_{F}(s).$ 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 corresponding groups of divisible elements