When does the rational torsion split off for finitely generated modules.
type: Item
instance of: scholarly article
title: When does the rational torsion split off for finitely generated modules.
zbMATH Open document ID: 1211.16024
author: Miodrag Cristian Iovanov
published in: Algebras and Representation Theory
publication date: 2009-09-01T00:00:00Z
review text: Let \(C\) be a coalgebra with counit over a field \(k\). Let \(\text{Rat}(M)\) denote the rational part of a \(C^*\)-module, where \(C^*\) denotes the convolution algebra associated to \(C\). The splitting problem for commutative and noncommutative rings is a classical question. \textit{C. Năstăsescu} and the reviewer [J. Algebra 281, No. 1, 144-149 (2004; Zbl 1078.16047)] have shown that if all \(C^*\)-modules splits, i.e. \(\text{Rat}(M)\) is a direct summand of \(M\), then the coalgebra \(C\) is finite dimensional. In this paper the author considers the more general problem of characterizing coalgebras with the splitting property only for finitely generated modules. It is shown that if \(C\) is a co-local (infinite dimensional) coalgebra, then the coalgebra \(C\) has the left finitely generated Rat-splitting property if and only if \(C\) is chain coalgebra. Moreover, if the base field \(K\) is algebraically closed or the coalgebra \(C\) is pointed, this is equivalent to \(C\) being isomorphic to the divided power coalgebra. Some classes of examples of coalgebras with the splitting property are constructed.
Mathematics Subject Classification ID: 16T15
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 18E40
zbMATH DE Number: 5600713
zbMATH Keywords: torsion theories
zbMATH Keywords: splitting property
zbMATH Keywords: coalgebras
zbMATH Keywords: comodules
zbMATH Keywords: rational modules
reviewed by: Blas Torrecillas
MaRDI profile type: Publication
full work available at URL: s10468-009-9144-7
OpenAlex ID: W2006008249
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DOI: 10.1007/S10468-009-9144-7
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description: scientific article; zbMATH DE number 5600713