MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

Is it true that if $\mbox{Ext}^{1}(P,M)=0$ for every finitely generated module $M$ then $P$ is projective? Or that if $\mbox{Ext}^{1}(M,Q)=0$ for every finitely generated module $M$ then $Q$ is injective?Does $\mbox{Ext}$ commute with direct limit?
Is it true that if $\mbox{Ext}^{1}(P,M)=0$ for every finitely generated module $M$ then $P$ is projective? Or that if $\mbox{Ext}^{1}(M,Q)=0$ for every finitely generated module $M$ then $Q$ is injective? Does $\mbox{Ext}$ commute with direct limit?