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Let $M$ be a representation of a quiver $Q=(V, E)$ by $R$-modules. By $M^{+}$ we mean a representation of $Q^{op}$ with $M^{+}(v)=\mathrm{Hom}(M(v), \frac{Q}{Z})$. One can easily see that there is anatural monomorphism $M\to M^{++}$ of representations of $Q$. Is it a pure monomorphism? That is, for any finitely presented representation $F$, can we prove that $ \mathrm{Hom}(F, M^{++})\rightarrow {\rm Hom}(F, \frac{M^{++}}{M})$ is an pimorphism?

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  • $\begingroup$ $Hom(F,-)$ is covariant. $\endgroup$ Oct 30, 2013 at 8:55
  • $\begingroup$ Does $Q$ have finitely many vertices? If so, doesn't this follow from the fact that $M\to M^{++}$ is a pure monomorphism for any $A$-module $M$ for any ring $A$, taking $A$ to be the path algebra $RQ$? $\endgroup$ Oct 31, 2013 at 14:32

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