Splitting in the K-theory localization sequence of number fields

TL;DR

This paper investigates the conditions under which the K-theory localization sequence for number fields splits, relating these conditions to class group coinvariants and providing explicit criteria and examples.

Contribution

It establishes necessary and sufficient conditions for the splitting of the K-theory localization sequence in number fields, involving coinvariants of twisted p-parts of class groups.

Findings

Conditions for sequence splitting depend on class group coinvariants.

Comparison with weaker cohomological conditions like WK^{et}_{2i}(F)=0.

Examples illustrating the theoretical criteria.

Abstract

Let p be a rational prime and let F be a number field. Then, for each i>0, there is a short exact localization sequence for K_{2i}(F). If p is odd or F is nonexceptional, we find necessary and sufficient conditions for this exact sequence to split: these conditions involve coinvariants of twisted p-parts of the p-class groups of certain subfields of the fields F(\mu_{p^n}) for n\in N. We also compare our conditions with the weaker condition WK^{et}_{2i}(F)=0 and give some example.