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In the paper Prime Representations from a Homological Perspective. The authors show that $Ext^{1}_{\hat{\mathcal{F}}}(V, V)$ is one-dimensional if and only if $V$ is prime for some modules $V$ of quantum affine algebras.

Therefore $Ext^{1}_{\hat{\mathcal{F}}}(V, V)$ is convient to discribe prime modules. Are there some other situations that $Ext^{1}_A(M, N)$ is used to descibe properties of A-modules $M, N$, where $A$ is some algebra? Are there some references about this? Thank you very much.

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Yes, $M$ is projective iff that Ext vanishes for all $N$, but this question is too elementary for a research forum, I believe. –  Fernando Muro May 5 '13 at 7:50
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