derived functor

Many natural functors are exact only halfway: they preserve one end of a short exact sequence but mangle the other. Derived functors are the precise, systematic measurement of that failure. Rather than mourn that a functor does not stay exact, one records the entire sequence of error terms it produces, organizing them into a tower of new functors that together restore an exact sequence — the long exact sequence — and encode deep structural information.

Concretely, suppose F is a left exact additive functor between module categories (or abelian categories with enough injectives). To compute its right derived functors R^n F at a module M, take an injective resolution 0 -> M -> I^0 -> I^1 -> ..., delete M, apply F to get a cochain complex F(I^0) -> F(I^1) -> ..., and set R^n F(M) = H^n of that complex. Then R^0 F = F, and a short exact sequence of modules yields a long exact sequence of derived functors. Dually, a right exact functor G has left derived functors L_n G computed from projective resolutions, with L_0 G = G.

The point is that the derived functors do not depend on the choice of resolution (any two are homotopy equivalent) and they vanish in positive degrees exactly when the functor is already exact. The classic examples are Ext (deriving Hom) and Tor (deriving tensor product). Sheaf cohomology, group cohomology, and Galois cohomology are all derived functors of suitable left exact functors, which is why they share the same formal machinery of long exact sequences and connecting maps.