2021/02/06 by Martino Lupini, Lupini, Martino
Mathematics · Computer Science · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2102.03477
The notion of phantom extension of order a given ordinal α 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 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∈ R. In this case, the phantom extensions of order α of a countable torsion module are precisely the pω( 1+α) -pure extensions introduced by Nunke in the 1960s. A module has projective length at most α if and only if it is a projective object with respect to the exact structure defined by phantom extensions of order α. We prove that a countable torsion module has projective length at most α if and only if it is reduced and has Ulm length at most 1+α, 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+α.