Derived functor

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. It was noted in various quite different settings that a short exact sequence often gives rise to a 'long exact sequence'. The concept of derived functors explains and clarifies many of these observations.The crucial assumption we need to make about our abelian category A is that it has enough injectives, meaning that for every object A in A there exists a monomorphism A → I where I is an injective object in A.If one starts with a covariant right-exact functor G, and the category A has enough projectives (i.e. for every object A of A there exists an epimorphism P → A where P is a projective object), then one can define analogously the left-derived functors LiG. For an object X of A we first construct a projective resolution of the formIf X {displaystyle X}   is a topological space, then the category S h ( X ) {displaystyle Sh(X)}   of all sheaves of abelian groups on X {displaystyle X}   is an abelian category with enough injectives. The functor Γ : S h ( X ) → A b {displaystyle Gamma :Sh(X) o Ab}   which assigns to each such sheaf F {displaystyle {mathcal {F}}}   the group Γ ( F ) := F ( X ) {displaystyle Gamma ({mathcal {F}}):={mathcal {F}}(X)}  of global sections is left exact, and the right derived functors are the sheaf cohomology functors, usually written as H i ( X , F ) {displaystyle H^{i}(X,{mathcal {F}})}  . Slightly more generally: if ( X , O X ) {displaystyle (X,{mathcal {O}}_{X})}   is a ringed space, then the category of all sheaves of O X {displaystyle {mathcal {O}}_{X}}  -modules is an abelian category with enough injectives, and we can again construct sheaf cohomology as the right derived functors of the global section functor.Derived functors and the long exact sequences are 'natural' in several technical senses.The more modern (and more general) approach to derived functors uses the language of derived categories.