The Iwasawa Algebra $\Omega_G$ and Its Dual Artin Coalgebra

TL;DR

This paper establishes a duality between the Iwasawa algebra of a compact p-adic Lie group and an associated artinian coalgebra, linking their derived categories of modules and comodules.

Contribution

It introduces a duality between the derived categories of modules over the Iwasawa algebra and comodules over an artinian coalgebra, expanding understanding of their algebraic structures.

Findings

$ ext{Omega}_G$ is the dual of an artinian coalgebra $C$.

A duality between derived categories of modules and comodules is established.

Provides new insights into the structure of Iwasawa algebras and their dual coalgebras.

Abstract

For any compact $p$-adic Lie group $G$, the Iwasawa algebra $\Omega_G$ over finite field $\mathbb{F}_p$ is a complete noetherian semilocal algebra. It is shown that $\Omega_G$ is the dual algebra of an artinian coalgebra $C$. We induce a duality between the derived category $\mathcal{D}^b_{fg}(_{\Omega_G}\mathcal{M})$ of bounded complexes of left $\Omega_G$-modules with finitely generated cohomology modules and the derived category $\mathcal{D}^b_{qf}(^C\mathcal{M})$ of bounded complexes of left $C$-comodules with quasi-finite cohomology comodules.