Cellular covers of divisible uniserial modules over valuation domains
type: Item
instance of: scholarly article
title: Cellular covers of divisible uniserial modules over valuation domains
author: László Fuchs
author: B. Goldsmith
author: Luigi Salce
author: Lutz Strüngmann
published in: Forum Mathematicum
publication date: 2024-10-01T00:00:00Z
review text: Let \(R\) be an integral domain with field of fractions \(Q\). A morphism \(\gamma\colon \Gamma \to M\) in \({\mathrm{Mod}\text{-}R}\) is a \textit{cellular cover} if it induces an isomorphism \(\Hom_R(\Gamma,\Gamma)\overset\cong\longrightarrow \Hom_R(\Gamma,M)\). Moreover, the following short exact sequence in \(\mathrm{Mod}\text{-}R\) \N\[\N0\to K:=\mathrm{Ker}(\gamma)\longrightarrow \Gamma \overset{\gamma}\longrightarrow M\to 0 \tag{$*$}\N\] \Nis said to be \textit{cellular} if \(\gamma\) is a cellular cover. Given a module \(M\), it is natural to look for some classification of its (surjective) cellular covers, either in general or requiring that \(\Gamma\) (and/or \(K\)) belong to suitable classes of modules. We refer to the well-written introduction of the paper for a detailed discussion of the state of the art on this kind of problems; the best known results are for cellular sequences of Abelian groups, when \(M\) is divisible.\N\NThe goal of the paper under review is to(at least partially) extend the results for cellular covers of divisible Abelian groups, to general (\(h\)-)divisible \(R\)-modules. The main results in this generality are the following (see Section~3 of the paper):\N\begin{itemize}\N\item[--] for \(M\) torsion-free and divisible, \((*)\) is cellular only if \(K=0\);\N\item[--] for \(M\) torsion and \(h\)-divisible, \((*)\) is cellular if and only if it is isomorphic to \(0\to \Hom_R(Q/R,M)\to \Hom_R(Q,M)\to M\to 0\), relating \(M\) with its Matlis dual.\N\end{itemize}\N\NStarting from Section~4, \(R\) is assumed to be a valuation domain, and \(M\) a torsion, divisible and uniserial \(R\)-module. In particular, in Sections~4 and 5, \(M\) (and, for the best results, also \(\Gamma\)) is assumed to be standard uniserial while, in Section~6, \(M\) (and, for the best results, also \(K\)) is assumed to be a ``clone of \(Q/R\)'' (these are a class of examples of uniserial divisible modules that are non-standard and, as a consequence, not \(h\)-divis…
reviewed by: Simone Virili
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 13C99
zbMATH DE Number: 7920529
zbMATH Keywords: clones
zbMATH Keywords: uniserial modules
zbMATH Keywords: divisible modules
zbMATH Keywords: cellular covers
MaRDI profile type: Publication
cites work: Q5483532
cites work: Q3400444
cites work: Localizations of torsion-free Abelian groups.
cites work: Co-local subgroups of Abelian groups. II.
cites work: Relative homological algebra. Vol. 1
cites work: Cellular covers of groups
cites work: On kernels of cellular covers
cites work: Arbitrarily large indecomposable divisible torsion modules over certain valuation domains
cites work: Abelian Groups
cites work: Cellular covers of Abelian groups.
cites work: Q3699812
cites work: Q4513600
cites work: On a non-vanishing Ext
cites work: Picard groups of some module categories
cites work: Approximations and endomorphism algebras of modules. Volume 1: Approximations. Volume 2: Predictions.
cites work: Cotorsion modules
cites work: Q4011056
cites work: Serial modules and endomorphism rings
cites work: Q4258039
DOI: 10.1515/FORUM-2023-0351
description: scientific article; zbMATH DE number 7920529