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?


closed as too localized by Andreas Blass, Mariano Suárez-Álvarez, Karl Schwede, Ralph, Qiaochu Yuan Feb 7 '12 at 18:51

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    $\begingroup$ 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 Choi Feb 7 '12 at 18:42
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    $\begingroup$ 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-Álvarez Feb 7 '12 at 18:43
  • $\begingroup$ @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. $\endgroup$ – Arold Feb 16 '12 at 19:50
  • $\begingroup$ Arold: for the definition of Ext without assuming enough projectives, see how Ext is defined in Maclane's book "Homology." $\endgroup$ – Charles Staats Feb 25 '12 at 17:50