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 (Theorem 1.10). 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 field, $\operatorname{ \sf MA}_{ \aleph_1}$ (Martin's Axiom for $\aleph_1$ many dense sets, which is a combinatorial axiom in set theory) implies that ${ \rm Ext}^1_{ KQ}(M_{ \omega_1},KQ)=0$ (Theorem 2.11).
Categories: math.RT math.LO