Thoughts on the reduced Whitehead group of the Iwasawa algebra

TL;DR

This paper discusses the properties of the reduced Whitehead group of Iwasawa algebras associated with certain Galois extensions, reducing the problem to cases with pro-l Galois groups and finite unramified extensions.

Contribution

It reduces the conjecture on the triviality of SK_1(QG) for Iwasawa algebras to simpler cases involving pro-l Galois groups and unramified extensions.

Findings

Reduces the conjecture to pro-l Galois groups case

Simplifies the problem to unramified coefficient extensions

Provides a framework for further investigation of SK_1(QG)

Abstract

Let l be an odd prime and K/k a Galois extension of totally real number fields with Galois group G such that K/k_\infty and k/Q are finite. We reduce the conjectured triviality of the reduced Whitehead group SK_1(QG) of the algebra QG=Quot(\Lambda G) with the Iwasawa algebra \Lambda G = Z_l[[G]] to the case of pro-l Galois groups G and finite unramified coefficient extensions.