余挠自由模的胞腔覆盖
Cellular covers of cotorsion-free modules
摘要：
本文改进了最近关于$R$ -模的胞腔覆盖的结果。细胞覆盖（有时称为共定域）出现在拓扑空间的同伦定域的背景下。它们与范畴论中的幂等余三元组、幂等余单子或余反射子有关。回想一下，如果$π$诱导同构$π_*: \Hom_R(G,G)\cong \Hom_R(G,H),$，其中$π_*(ϕ)= πϕ$用于每个$ϕ\in \Hom_R(G,G)$（其中映射作用于左侧），则$R$ -模块$π: G\to H$的同态称为$H$上的{ \it蜂窝覆盖}。一方面，我们证明了每个秩为$κ<\Cont$的无压缩$R$ -模都可以实现为某个蜂窝覆盖$G\to H$的核，其中$G$的秩为$3κ+1$（或3，如果$κ=1$）。证明是基于角落的经典思想，如何构建扭自由阿贝尔群规定可数自同态环。这补充了巴克纳-杜加斯\cite{BD}的结果。另一方面，我们证明了每一个满足刚性条件的无负扭$R$ -模$H$都允许任意大的胞腔覆盖$G\to H$。这改进了Fuchs-Göbel \cite{FG}和Farjoun-Göbel-Segev-Shelah \cite{FGSS07}的结果。
英文摘要：
In this paper we improve recent results dealing with cellular covers of $R$-modules. Cellular covers (sometimes called co-localizations) come up in the context of homotopical localization of topological spaces. They are related to idempotent cotriples, idempotent comonads or coreflectors in category theory. Recall that a homomorphism of $R$-modules $π: G\to H$ is called a {\it cellular cover} over $H$ if $π$ induces an isomorphism $π_*: \Hom_R(G,G)\cong \Hom_R(G,H),$ where $π_*(ϕ)= πϕ$ for each $ϕ\in \Hom_R(G,G)$ (where maps are acting on the left). On the one hand, we show that every cotorsion-free $R$-module of rank $κ<\Cont$ is realizable as the kernel of some cellular cover $G\to H$ where the rank of $G$ is $3κ+1$ (or 3, if $κ=1$). The proof is based on Corner's classical idea of how to construct torsion-free abelian groups with prescribed countable endomorphism rings. This complements results by Buckner--Dugas \cite{BD}. On the other hand, we prove that every cotorsion-free $R$-module $H$ that satisfies some rigid conditions admits arbitrarily large cellular covers $G\to H$. This improves results by Fuchs-Göbel \cite{FG} and Farjoun-Göbel-Segev-Shelah \cite{FGSS07}.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
