K_1 of certain Iwasawa algebras, after Kakde

TL;DR

This paper provides a detailed, axiomatic exposition of K_1 of certain Iwasawa algebras, refining Kakde's work by clarifying group requirements and utilizing a conceptual theory of localization to simplify arguments.

Contribution

It offers a more axiomatic approach to K_1 of Iwasawa algebras and incorporates a conceptual localization theory to streamline existing proofs.

Findings

Clarified conditions on p-adic Lie groups for K_1 analysis

Simplified arguments using localization theory

Extended Kakde's results with a more axiomatic framework

Abstract

This paper contains a detailed exposition of the content of section five in Kakde's paper arXiv:1008.0142. We proceed in a slightly more axiomatic way to pin down the exact requirements on the $p$-adic Lie group under consideration. We also make use of our conceptual theory of the completed localization of an Iwasawa algebra as developed in arXiv:0711.2669. This simplifies some of the arguments. Otherwise, with the exception of the notation at certain places, we follow Kakde's paper.