Given an abelian category C, we can form the Yoneda extensions $YExt^i(X,Y)$ to the equivalent classes of $i$-extensions of X by Y. 
Given any abelian category C, we can always formulate the derived category D(C), and define $Ext^i\_C(X,Y)$ to be $Hom\_{D(C)}(X,Y[i])$.


Now we can naturally associate a Yoneda $i$-extension
$$0\rightarrow Y\rightarrow Z\_{i-1}\rightarrow \cdots\rightarrow Z\_0\rightarrow X\rightarrow 0$$
 of X by Y the element $Y[i]\leftarrow[Y\rightarrow Z\_{i-1}\rightarrow \cdots\rightarrow Z\_0]\rightarrow X$ of $Hom\_{D(C)}(X, Y[i])$.


I am wondering when this map is inj (resp. surjective, bijective)?


We know this is an isomorphism under the assumption that C admits enough injective objects or projective objects.



group of Yoneda extensions and the EXT groups defined via derived category
It is true in general and Verdier did the computation in his thesis Des Catégories Dérivées des Catégories Abéliennes Chap III Sect 3 ... the thesis is on line!


