Projective length, phantom extensions, and the structure of torsion modules

TL;DR

This paper characterizes phantom extensions of torsion modules over countable Dedekind domains, linking them to pure extensions and projective length, and provides a structural description involving Ulm length and colimits.

Contribution

It offers a detailed characterization of phantom extensions of torsion modules, connecting them to pure extensions, projective length, and module colimits over well-founded forests.

Findings

Phantom extensions of torsion modules correspond to specific pure extensions.

A module's projective length is characterized by its Ulm length and structural properties.

Countable torsion modules can be described as colimits over well-founded forests.

Abstract

The notion of phantom extension of order a given ordinal $\alpha $ has been introduced in collaboration with Casarosa, as an algebraic analogue of the order of a phantom map in topology, to study the structure of flat modules. In this companion paper we characterize phantom extension of \emph{torsion} modules over a countable Dedekind domain $R$. After localizing, one can assume that $R$ is a discrete valuation domain with maximal ideal generated by $p\in R$. In this case, the phantom extensions of order $\alpha $ of a countable torsion module are precisely the $p^{\omega \left( 1+\alpha\right) }$-pure extensions introduced by Nunke in the 1960s. A module has projective length at most $\alpha $ if and only if it is a projective object with respect to the exact structure defined by phantom extensions of order $\alpha $. We prove that a countable torsion module has projective length at most $\alpha $ if and only if it is reduced and has Ulm length at most $1+\alpha $, if and only if it is the colimit of a presheaf of finite torsion modules over a countable well-founded forest of rank at most $1+\alpha $.