Derived Functor - Construction and First Properties

Construction and First Properties

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.

The right derived functors of the covariant left-exact functor F : A → B are then defined as follows. Start with an object X of A. Because there are enough injectives, we can construct a long exact sequence of the form

where the I i are all injective (this is known as an injective resolution of X). Applying the functor F to this sequence, and chopping off the first term, we obtain the chain complex

Note: this is in general not an exact sequence anymore. But we can compute its homology at the i-th spot (the kernel of the map from F(Ii) modulo the image of the map to F(Ii)); we call the result RiF(X). Of course, various things have to be checked: the end result does not depend on the given injective resolution of X, and any morphism X → Y naturally yields a morphism RiF(X) → RiF(Y), so that we indeed obtain a functor. Note that left exactness means that 0 →F(X) → F(I0) → F(I1) is exact, so R0F(X) = F(X), so we only get something interesting for i>0.

(Technically, to produce well-defined derivatives of F, we would have to fix an injective resolution for every object of A. This choice of injective resolutions then yields functors RiF. Different choices of resolutions yield naturally isomorphic functors, so in the end the choice doesn't really matter.)

The above-mentioned property of turning short exact sequences into long exact sequences is a consequence of the snake lemma. This tell us that the collection of derived functors is a δ-functor.

If X is itself injective, then we can choose the injective resolution 0 → X → X → 0, and we obtain that RiF(X) = 0 for all i ≥ 1. In practice, this fact, together with the long exact sequence property, is often used to compute the values of right derived functors.

An equivalent way to compute RiF(X) is the following: take an injective resolution of X as above, and let Ki be the image of the map Ii-1→Ii (for i=0, define Ii-1=0), which is the same as the kernel of Ii→Ii+1. Let φ_i : Ii-1→Ki be the corresponding surjective map. Then RiF(X) is the cokernel of F(φ_i).