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| Natura: | Preprint |
| Pubblicazione: |
2021
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2102.03477 |
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Sommario:
- 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 $1+α$.