Ext functor

Summary

The Ext functors $\operatorname{Ext}^n_R(M, N)$ are the right derived functors of $\Hom_R(-, N)$. They measure how far $\Hom$ is from being exact. The group $\operatorname{Ext}^1_R(M, N)$ classifies the extensions of $M$ by $N$, and the vanishing of Ext detects projective modules, injective modules and the projective dimension.