Derived functor

Summary

The derived functors of a left exact functor $F$ are functors $R^n F$, for $n \ge 0$, with $R^0 F = F$. They are computed by applying $F$ to an injective resolution and taking cohomology. They turn a short exact sequence into a long exact sequence, and so measure how far $F$ is from being exact. Ext, Tor, group cohomology and sheaf cohomology are all derived functors.