Absolutely pure representations of quivers.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1310.16014
DOI: 10.4134/JKMS.2014.51.6.1177
author: Mansour Aghasi
author: Hamidreza Nemati
published in: Journal of the Korean Mathematical Society
publication date: 2014-11-14T00:00:00Z
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 6369562
zbMATH Keywords: quiver representations
zbMATH Keywords: absolutely pure representations
zbMATH Keywords: flat representations
zbMATH Keywords: covering classes
MaRDI profile type: Publication
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title: Absolutely pure representations of quivers.
review text: Let \(Q\) be a quiver and let \(R\) be a commutative ring. The category of representations of the quiver \(Q\) by \(R\)-modules is denoted by \(\mathrm{Rep}(Q,R)\). The authors show that for coherent rings the class of all componentwise absolutely pure representations is a covering class. Let \(X\) be a representation of \(Q\) by \(R\)-modules, the representation \(X^+\) of \(Q^{op}\) by \(R\)-modules is given by the following: for any vertex \(v\), \(X^+(v)=\Hom_{\mathbb Z}(X(v),\mathbb Q/\mathbb Z)\), and for any arrow \(a\colon v\to w\), \(X^+(a)\colon X^+(w)\to X^+(v)\). \textit{E. Hosseini} [in Bull. Korean Math. Soc. 50, No. 2, 389-398 (2013; Zbl 1288.16017)] introduced the notion of purity for representations: a monomorphism of representations \(f\colon X\to Y\) is called pure if \(f^+\colon Y^+\to X^+\) is a split epimorphism of representations of \(Q^{op}\).NEWLINENEWLINE In this paper, the authors study absolutely pure representations, a representation \(X\) is absolutely pure if every monomorphism \(0\to X\to Y\) is pure. If \(X\) is an absolutely pure representation then (i) \(X(v)\) is absolutely pure and (ii) For any vertex \(v\) the morphism \(X(v)\to\prod_{s(a)=v}X(t(a))\) induced by \(X(v)\to X(t(a))\) is a pure epimorphism, here \(s\) and \(t\) denote the initial and terminal applications for arrows. The converse is true if \(R\) is a coherent ring and \(Q\) a rooted quiver. Finally, it is shown that this class of representations is also covering. This extends the result for modules by \textit{K. Pinzon} [Commun. Algebra 36, No. 6, 2186-2194 (2008; Zbl 1162.16003)].
reviewed by: Blas Torrecillas
description: scientific article; zbMATH DE number 6369562