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One can define $\mathrm{Ext}^n(M,N)$ (where $M,N$ are $R$-modules) in two ways, either by taking an injective resolution of $N$ and applying $\mathrm{Hom}(M,-)$or by taking a projective resolution and applying $\mathrm{Hom}(-,N)$.

I'm struggling to remember which is which and I'm sure there is a(n obvious?) reason. Why does it have to be projective + $\mathrm{Hom}(-,N)$ or injective + $\mathrm{Hom}(M,-)$ but cannot be projective + $\mathrm{Hom}(M,-)$?

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    @MattN. By definition, projective modules are the ones that have nice properties when you look at functions out of them, and injectives are the ones that have nice properties when you look at functions into them. That's one way to remember which side to use.2012-07-18

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