= Long exact sequence of Ext groups
Applying the contravariant <Hom functor> $\operatorname{Hom}_R(-,W)$ to $0\to I\to X\to C\to0$ gives
$$
0\to\operatorname{Hom}(C,W)\to\operatorname{Hom}(X,W)\to\operatorname{Hom}(I,W)\to\operatorname{Ext}^1(C,W)\to\operatorname{Ext}^1(X,W)\to\operatorname{Ext}^1(I,W)\to\operatorname{Ext}^2(C,W)\to\cdots.
$$
The connecting map sends $I\to W$ to its <pushout of a module extension>. Its vanishing means that map extends to $X\to W$. For a <hereditary ring>, $\operatorname{Ext}^2(C,W)=0$, making the indicated restriction on first <extension groups> surjective. The covariant variable has its corresponding long exact sequence.
Back to article page
