Interpretation functors which are full on pure-injective modules with applications to $R$-torsion-free modules over $R$-orders

TL;DR

This paper studies interpretation functors that are full on pure-injective modules, applying the results to describe torsion-free modules over orders and introducing pseudogeneric modules to extend concepts from finite-dimensional algebra.

Contribution

It proves that certain interpretation functors full on a subcategory are also full on pure-injective modules and introduces pseudogeneric modules for orders.

Findings

Interpretation functors full on a subcategory are full on pure-injective modules.

Application to describe torsion-free modules over tame Bäckström orders.

Introduction of pseudogeneric modules for modules over orders.

Abstract

Let $R,S$ be rings, $\mathcal{X}\subseteq \text{mod}$-$R$ a covariantly finite subcategory, $\mathcal{C}$ the smallest definable subcategory of $\text{Mod}$-$R$ containing $\mathcal{X}$ and $\mathcal{D}$ a definable subcategory of $\text{Mod}$-$S$. We show that if $I:\mathcal{C}\rightarrow \mathcal{D}$ is an interpretation functor such that $I\mathcal{X}\subseteq \text{mod}$-$S$ and whose restriction to $\mathcal{X}$ is full then $I$ is full on pure-injective modules. We apply this theorem to an extension of a functor introduced by Ringel and Roggenkamp which, in particular, allows us to describe the torsion-free part of the Ziegler spectra of tame B\"ackstr\"om orders. We also introduce the notion of a pseudogeneric module over an order which is intended to play the same role for lattices over orders as generic modules do for finite-dimensional modules over finite-dimensional algebras.