LAUSR

In this paper it is proved that, when $Q$ is a quiver that admits some closure, for any algebraically closed field $K$ and any finite dimensional $K$-linear representation $\mathcal{X}$ of $Q$, if ${\rm Ext}^1_{KQ}(\mathcal{X},KQ)=0$ then $\mathcal{X}$ is projective. In contrast, we show that if $Q$ is a specific quiver of the type above, then there is an infinitely generated non projective $KQ$-module $M_{\omega_1}$ such that, when $K$ is a countable algebraically closed field, Martin's Axiom (which is a combinatorial axiom in set theory) implies that ${\rm Ext}^1_{KQ}(M_{\omega_1},KQ)=0$.