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 Ext commute with direct limit?
For the first question you already have had an answer in [Is it true that if $\operatorname{Ext}^{1}\_{A}(P,A/I)=0 $ for all $ I$ then $P$ is projective?](https://mathoverflow.net/questions/25687/projective-module/25698) if $\mathrm{Ext}^1\_{\mathbb Z}(P,M)=0$, then it depends on the axioms of set-theory whether the conclusion is true or not. The answer to the second question is yes, it is one of the basic characterisation of injective modules that $\mathrm{Ext}^1(A/I,Q)=0$ for all ideals $I$ iff $Q$ is injective. As for the question in your title, the answer should be no for the second variable (irrespective of the axioms of set theory, but I am too lazy to try to come up with an example). For the first variable things are a little bit more interesting: If $M$ is the direct limit of ${M\_\alpha}$, then we have a spectral sequence with $E\_2$-term $lim^i\mathrm{Ext}^j(M\_{\alpha},Q)$ (*"lim" means inverse limit, there is some strange problem with using "varprojlim" which sometimes works and sometimes doesn't*) and converging to $\mathrm{Ext}^{i+j}(M,Q)$. Somewhat strangely this spectral sequence does not seem to formally give the above characterisation of injective modules as there is a potential $lim^1\mathrm{Hom}(M\_{\alpha},Q)$ contribution. 


