DerivedFunktoren

Derived Functors are a tool in homological algebra that extend the notion of a functor to measure how far it is from being exact. They form a way to assign, to each object and each degree, a new abelian group or module that captures the failure of exactness in a systematic, functorial way.

Construction. Let A be an abelian category with enough injectives and F: A → B a left exact functor to another abelian category B. The right derived functors R^nF are defined by taking an injective resolution I^• of an object A and setting R^nF(A) = H^n(F(I^•)). Dually, if F is right exact, its left derived functors L_nF are defined using a projective resolution P_• of A by L_nF(A) = H_n(F(P_•)). For a short exact sequence 0 → A' → A → A'' → 0 there are natural long exact sequences relating the R^nF or L_nF.

Key examples. Ext and Tor arise as derived functors: Ext^n(A,B) ≅ R^n Hom(A,B), and Tor_n^R(A,B) can be computed as L_n(- ⊗^L_R B). In many settings, R^0F ≅ F when F is left exact, and L_0F ≅ F when F is right exact.

Abstract viewpoint. Derived functors can be defined in the derived category as total derived functors, and they are well-defined up to natural isomorphism, functorial in their arguments for bifunctors.

Applications. They underpin cohomology theories, facilitate computation of invariants, and appear in spectral sequences and long exact sequences that relate objects across exact sequences.