[Paper Review] K_1 of some Iwasawa algebras

Mahesh Kakde|arXiv (Cornell University)|May 22, 2010

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper computes the Whitehead group $K_1$ of Iwasawa algebras for one-dimensional compact $p$-adic Lie groups $\mathcal{G}$ with a quotient isomorphic to $\mathbb{Z}_p$, using the integral logarithm of Oliver and Taylor. It establishes an exact sequence relating $K_1(\Lambda_O(\mathcal{G}))$ to the group of units and abelianization of $\mathcal{G}$, and proves that the localization at the Ore set $S$ preserves key structural properties, generalizing earlier results by Kato and Hara.

ABSTRACT

Noncommutative Iwasawa theory has created a lot of interest in Whitehead groups of Iwasawa algebras of compact p-adic Lie groups with a quotient isomorphic to the additive group of p-adic integers. In this paper we compute Whitehead groups of Iwasawa algebra of a pro-p compact p-adic Lie group of dimension one. We also give results on Whitehead groups of the localisation of such Iwasawa algebras at the canonical Ore set defined by Coates, Fukaya, Kato, Sujatha and Venjakob.

Motivation & Objective

• To describe the Whitehead group $K_1(\Lambda_O(\mathcal{G}))$ for compact $p$-adic Lie groups $\mathcal{G}$ of dimension one with $\mathcal{G}/H \cong \mathbb{Z}_p$.
• To generalize results of Kato, Hara, and Kakde on $K_1$ of Iwasawa algebras to this class of groups.
• To analyze the structure of the localized Iwasawa algebra $\Lambda_O(\mathcal{G})_S$ via the Ore set $S$ defined by Coates et al.
• To establish an exact sequence involving $K_1$, the unit group, and the abelianization of $\mathcal{G}$, using the integral logarithm as a key tool.

Proposed method

• Represent the Iwasawa algebra $\Lambda_O(\mathcal{G})$ as a twisted group ring $\Lambda_O(\Gamma^{p^e})[G]^\tau$ using a symmetric 2-cocycle $\tau$.
• Define the Ore set $S$ as the set of elements $f$ such that $\Lambda_O(\mathcal{G})/\Lambda_O(\mathcal{G})f$ is a finitely generated $O$-module.
• Construct the integral logarithm map $\mathcal{L}: \Phi^G \to \psi^G$ using the logarithm of elements in the twisted group ring and the image of the $\beta^G$-map.
• Use the commutative diagram involving $\theta^G$, $\beta^G$, and $\mathcal{L}$ to relate $K_1(\Lambda_O(\mathcal{G}))$ to the unit group and abelianization.
• Prove that the localization $\Lambda_O(\mathcal{G})_T$ at $T = \Lambda_O(\Gamma^{p^e}) \setminus p\Lambda_O(\Gamma^{p^e})$ is isomorphic to $\Lambda_O(\mathcal{G})_S$, enabling the use of localization techniques.
• Apply the five lemma to the commutative diagram to deduce the exactness of the sequence involving $K_1$ and the structure of $\Phi^G$.

Experimental results

Research questions

• RQ1 What is the structure of $K_1(\Lambda_O(\mathcal{G}))$ for a one-dimensional compact $p$-adic Lie group $\mathcal{G}$ with $\mathcal{G}/H \cong \mathbb{Z}_p$?
• RQ2 How does the integral logarithm of Oliver and Taylor relate to the $K_1$-group of the Iwasawa algebra?
• RQ3 What is the image of the map $\theta^G: K_1(\Lambda_O(\mathcal{G})) \to \Lambda_O(\Gamma^{p^e})[\mathrm{Conj}(G)]^\tau$?
• RQ4 How does localization at the Ore set $S$ affect the structure of $K_1$ and the unit group of the Iwasawa algebra?
• RQ5 Is the localization map $\Lambda_O(\mathcal{G})_T \to \Lambda_O(\mathcal{G})_S$ an isomorphism for the relevant Ore set $T$?

Key findings

• The map $\theta^G: K_1(\Lambda_O(\mathcal{G})) \to \Lambda_O(\Gamma^{p^e})[\mathrm{Conj}(G)]^\tau$ is surjective onto the image of $\beta^G$, and the kernel is isomorphic to $\mu(O) \times \mathcal{G}^{ab}$.
• The exact sequence $1 \to \mu(O) \times \mathcal{G}^{ab} \to \Phi^G \xrightarrow{\mathcal{L}} \psi^G \xrightarrow{\omega} \langle(-1)^{p-1}\rangle \times \mathcal{G}^{ab} \to 1$ holds, with $\mathcal{L}$ defined via the integral logarithm.
• The localization $\Lambda_O(\mathcal{G})_T$ at $T = \Lambda_O(\Gamma^{p^e}) \setminus p\Lambda_O(\Gamma^{p^e})$ is isomorphic to $\Lambda_O(\mathcal{G})_S$, so $S$-localization is equivalent to $T$-localization.
• The image of $\theta^G_S$ in $\Phi^G_S$ is contained in $\Phi^G_S$, and $\Phi^G_S \cap \prod_{P \in C(G)} \Lambda_O(U_P)^\times = \mathrm{Im}(\theta^G)$, showing compatibility with localization.
• The five lemma is applied to a commutative diagram involving $K_1$, $\theta^G$, $\beta^G$, and $\mathcal{L}$ to prove that $K_1(\Lambda_O(\mathcal{G}))$ maps isomorphically onto the kernel of $\omega \circ \mathcal{L}$.
• The result generalizes earlier work by Kato and Hara, extending the $K_1$-computation to all one-dimensional $p$-adic Lie groups with $\mathcal{G}/H \cong \mathbb{Z}_p$.