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

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1. Verfasser: Lupini, Martino
Format: Preprint
Veröffentlicht: 2021
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author Lupini, Martino
author_facet Lupini, Martino
contents 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 \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 $α$ of a countable torsion module are precisely the $p^{ω\left( 1+α\right) }$-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 $1+α$.
format Preprint
id arxiv_https___arxiv_org_abs_2102_03477
institution arXiv
publishDate 2021
record_format arxiv
spellingShingle Projective length, phantom extensions, and the structure of torsion modules
Lupini, Martino
Group Theory
Commutative Algebra
Logic
20K10, 20K35, 54H05 (Primary), 20K40, 20K45 (Secondary)
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 \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 $α$ of a countable torsion module are precisely the $p^{ω\left( 1+α\right) }$-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 $1+α$.
title Projective length, phantom extensions, and the structure of torsion modules
topic Group Theory
Commutative Algebra
Logic
20K10, 20K35, 54H05 (Primary), 20K40, 20K45 (Secondary)
url https://arxiv.org/abs/2102.03477