Absolutely pure representations of quivers. (Q2929756): Difference between revisions

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)].