$K_0$-invariance of the completely faithful property of Iwasawa modules

TL;DR

This paper proves that the property of being completely faithful for Iwasawa modules remains invariant under certain algebraic quotients and extends existing theorems to broader classes of $p$-adic Lie groups.

Contribution

It establishes $K_0$-invariance of the completely faithful property and generalizes a theorem of Ardakov to more general $p$-adic Lie groups.

Findings

Invariance of complete faithfulness under Grothendieck group classes

Extension of Ardakov's theorem to broader $p$-adic groups

Identification of conditions for $K_0$-invariance in Iwasawa modules

Abstract

Let $H$ be a compact $p$-adic analytic group without torsion element, whose Lie algebra is split semisimple and $\mathfrak{N}_H(G)$ be the full subcategory of the category of finitely generated modules over the Iwasawa algebra $\Lambda_G$ that are also finitely generated as $\Lambda_H$-modules, where $G = \mathbb{Z}_{p} \times H$. We show that if the class of a module $N$ in the Grothendieck group of $\mathfrak{N}_H(G)$ equals to the class of a completely faithful module, then $q(N)$ is also completely faithful, where $q(N)$ denotes the image of $N$ via the quotient functor modulo the full subcategory of pseudonull modules. We also generalize a Theorem of Konstantin Ardakov characterizing the completely faithful property to the case of more general $p$-adic Lie groups.