Let $P$ be a module in an abelian category $\mathcal C$ satisfying $Ext(P,M)=0$ for all modules $M \in \mathcal C$.
Can we conclude that $P$ is projective? Any reference for this?
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2$\begingroup$ Hint: long exact sequence for Ext. (your question implicitly presumes we have enough projectives to define Ext, unless you are using the Yoneda definition.) Perhaps you could give some background context for your question? are you working through parts of a book? $\endgroup$– Yemon ChoiFeb 7, 2012 at 18:42
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3$\begingroup$ You can use the usual characterization of projectivity in terms of the lifting property (see en.wikipedia.org/wiki/Projective_module#Lifting_property) and the exact sequence for the $\hom$ and $\mathrm{Ext}$ functors to prove this. This is proved in more or less any textbook on homological algebra. $\endgroup$– Mariano Suárez-ÁlvarezFeb 7, 2012 at 18:43
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$\begingroup$ @Yemon Choi: In my case I do have enough projectives. However, I am interested about your comment. Couldn't I talk about Ext if I don't have enough projectives? Why? I will appreciate an answer. $\endgroup$– AroldFeb 16, 2012 at 19:50
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$\begingroup$ Arold: for the definition of Ext without assuming enough projectives, see how Ext is defined in Maclane's book "Homology." $\endgroup$– Charles StaatsFeb 25, 2012 at 17:50
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