射影长度、幻影扩张与挠模的结构
Projective length, phantom extensions, and the structure of torsion modules
AI总结：
本文刻画了可数戴德金整环上挠模的幻影扩张，证明射影长度至多$\alpha$等价于约化且Ulm长度至多$1+\alpha$，并给出有限挠模预层的余极限刻画。
摘要：
给定序数 $\alpha$ 的幻影扩张的概念是与 Casarosa 合作引入的，作为拓扑学中幻影映射阶数的代数类比，用于研究平坦模的结构。在这篇配套论文中，我们刻画了可数戴德金整环 $R$ 上挠模的幻影扩张。局部化之后，可以假设 $R$ 是离散赋值整环，其极大理想由 $p\in R$ 生成。在这种情况下，可数挠模的 $\alpha$ 阶幻影扩张恰好是 Nunke 在 20 世纪 60 年代引入的 $p^{\omega\left( 1+\alpha\right) }$-纯扩张。一个模的射影长度至多为 $\alpha$，当且仅当它是相对于由 $\alpha$ 阶幻影扩张定义的精确结构的射影对象。我们证明，一个可数挠模的射影长度至多为 $\alpha$，当且仅当它是约化的且 Ulm 长度至多为 $1+\alpha$，当且仅当它是秩至多为 $1+\alpha$ 的可数良基森林上的有限挠模预层的余极限。
英文摘要：
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 $1+α$.
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