Maranda's Theorem for Pure-Injective Modules and Duality

TL;DR

This paper extends Maranda's theorem to pure-injective modules over orders in separable algebras, linking module reductions to indecomposability and spectrum topology, with applications to pure-injective hulls and module classification.

Contribution

It generalizes Maranda's theorem to a broader class of modules and establishes a duality between spectra of torsion-free modules over orders.

Findings

Extension of Maranda's theorem to pure-injective modules

Isomorphism of spectra lattices for left and right Ziegler spectra

Characterization of pure-injective hulls of torsion-free modules

Abstract

Let $R$ be a discrete valuation domain with field of fractions $Q$ and maximal ideal generated by $\pi$. Let $\Lambda$ be an $R$-order such that $Q\Lambda$ is a separable $Q$-algebra. Maranda showed that there exists $k\in\mathbb{N}$ such that for all $\Lambda$-lattices $L$ and $M$, if $L/L\pi^k\simeq M/M\pi^k$ then $L\simeq M$. Moreover, if $R$ is complete and $L$ is an indecomposable $\Lambda$-lattice, then $L/L\pi^k$ is also indecomposable. We extend Maranda's theorem to the class of $R$-reduced $R$-torsion-free pure-injective $\Lambda$-modules. As an application of this extension, we show that if $\Lambda$ is an order over a Dedekind domain $R$ with field of fractions $Q$ such that $Q\Lambda$ is separable then the lattice of open subsets of the $R$-torsion-free part of the right Ziegler spectrum of $\Lambda$ is isomorphic to the lattice of open subsets of the $R$-torsion-free part of the left Ziegler spectrum of $\Lambda$. Finally, with $k$ as in Maranda's theorem, we show that if $M$ is $R$-torsion-free and $H(M)$ is the pure-injective hull of $M$ then $H(M)/H(M)\pi^k$ is the pure-injective hull of $M/M\pi^k$. We use this result to give a characterisation of $R$-torsion-free pure-injective $\Lambda$-modules and describe the pure-injective hulls of certain $R$-torsion-free $\Lambda$-modules.