Projective length, phantom extensions, and the structure of torsion modules
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866915356263579648 |
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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 |
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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 |