On the splitting of the exact sequence, relating the wild and tame kernels

Abstract

Let k be a number field. For an odd prime p and an integer i≥2, let Шe´t2(k,Zp(i)) denote the étale wild kernel of k (corresponding to p and i). Then Шe´t2(k,Zp(i)) is contained in the finite group He´t2(ok′,Zp(i)), where ok′ is the ring of p-integers of k. We give conditions for the inclusion Шe´t2(k,Zp(i))⊆He´t2(ok′,Zp(i)) to split. We analyze this problem using Iwasawa theory. In particular we relate this splitting problem to the triviality of two invariants, namely the asymptotic kernels of the Galois descent and codescent for class groups along the cyclotomic tower of k. We illustrate our results in both split and non-split cases for quadratic number fields.