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

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 $1+\alpha$.