The integral logarithm in Iwasawa theory: an exercise

TL;DR

This paper investigates the structure of the unit group and the properties of the integral logarithm in the context of non-commutative Iwasawa theory, focusing on the algebraic objects associated with finite abelian l-groups.

Contribution

It explicitly determines the unit group of a localized completed group algebra and analyzes the kernel and cokernel of the integral logarithm in this setting.

Findings

Unit group of mbda_wedge[H] is explicitly characterized.

Kernel and cokernel of the integral logarithm are identified.

Results contribute to understanding algebraic structures in non-commutative Iwasawa theory.

Abstract

Let $l$ be an odd prime number and $H$ a finite abelian $l$-group. We determine the unit group of $\Lambda_\wedge[H]$ (the completion of the localization at $l$ of $\Bbb{Z}_l[[T]][H]$) as well as the kernel and cokernel of the integral logarithm $L:\Lambda_\wedge[H]^\times\to \Lambda_\wedge[H]$, which appears in non-commutative Iwasawa theory.