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

TL;DR

This paper extends the concept of phantom maps to triangulated categories, introduces phantom extensions of various orders for flat modules over Dedekind domains, and characterizes their structure and projective resolutions.

Contribution

It develops a new framework for phantom extensions in flat modules, establishes a dichotomy theorem, and provides a structure theorem analogous to Ulm classification.

Findings

Either all extensions are trivial or modules have arbitrarily high order phantom extensions.

Introduces hereditary exact structures based on phantom extensions.

Characterizes modules with bounded projective length as colimits over well-founded forests.

Abstract

We consider the natural generalization of the notion of the order of a phantom map from the topological setting to triangulated categories. When applied to the derived category of the category of countable flat modules over a countable Dedekind domain, this yields a notion of\emph{\ phantom extension} of order $\alpha <\omega _{1}$. We provide a complexity-theoretic characterization of the module $\mathrm{Ph}% ^{\alpha }\mathrm{Ext}\left( C,A\right) $ of phantom extensions of order $\alpha $ with respect to the structure of \emph{phantom Polish module} on $\mathrm{Ext}\left( C,A\right) $ obtained by considering it as an object of the left heart of the quasi-abelian category of Polish modules. We use this characterization to prove the following Dichotomy Theorem: either all the extensions of a countable flat module $A$ are trivial (which happens precisely when $A$ is divisible) or $A$ has phantom extensions of arbitrarily high order. By producing canonical phantom projective resolutions of order $\alpha $, we prove that phantom extensions of order $\alpha $ define on the category of countable flat modules an exact structure $\mathcal{E}_{\alpha }$ that is hereditary with enough projectives, and the functor $\mathrm{Ph}^{\alpha }% \mathrm{Ext}$ is the derived functor of $\mathrm{Hom}$ with respect to $\mathcal{E}_{\alpha }$. We prove a Structure Theorem characterizing the objects of the class $\mathcal{P}_{\alpha }$ of countable flat modules that have \emph{projective length at most }$\alpha $ (i.e., are $\mathcal{E}_{\alpha }$-projective) as the direct summands of colimits of presheaves of finite flat modules over well-founded forests of rank $% 1+\alpha $ regarded as ordered sets. This can be seen as the first analogue in the flat case of the classical Ulm Classification Theorem for torsion modules.