Derived functor
From Academic Kids

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 is not related in any way to the derivatives of calculus.)
Contents 
Motivation
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.
Suppose we are given a covariant left exact functor F : A → B between two abelian categories A and B. If
 <math>0\to A \to B \to C \to 0<math>
is a short exact sequence in A, then applying F yields the exact sequence
 <math>0\to F(A)\to F(B)\to F(C)<math>
and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is illposed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if A is "nice" enough) there is one "correct", "canonical", "natural" way of doing so, given by the right derived functors of F. For every i≥1, there is a functor R^{i}F: A → B, and the above sequence continues like so:
 <math>0\to F(A)\to F(B)\to F(C)\to R^1F(A) \to R^1F(B) \to R^1F(C)\to R^2F(A)\to \cdots<math>
From this we see that F is an exact functor if and only if R^{1}F = 0; so in a sense the right derived functors of F measure "how far" F is from being exact.
If the object A in the above short exact sequence is injective, then the sequence splits. Applying a functor to a split sequence results in a split sequence, and so R^{1}F(A) = 0. Right derived functors are zero on injectives: this is the motivation for the construction given below.
Construction and first properties
The crucial assumption we need to make about our abelian category A is that it have 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 leftexact functor F : A → B is 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
 <math>0\to X\to I^0\to I^1\to I^2\to\cdots<math>
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
 <math>0\to F(I^0)\to F(I^1) \to F(I^2) \to\cdots<math>
Note: this is in general not an exact sequence anymore. But we can compute its homology at the ith spot (kernel of the map F(I^{i})→F(I^{i+1}) modulo image of the map F(I^{i1})→F(I^{i})); the result for i≥1 we call R^{i}F(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 R^{i}F(X) → R^{i}F(Y), so that we indeed obtain a functor.
(Technically, to produce welldefined derivatives of F, we would have to fix an injective resolution for every object of A. This choice of injective resolutions then yields functors R^{i}F. Different choices of resolutions yield naturally isomorphic functors, so in the end the choice doesn't really matter.)
The abovementioned property of turning short exact sequences into long exact sequences is a consequence of the snake lemma.
If X is itself injective, then we can choose the injective resolution 0 → X → X → 0, and we obtain that R^{i}F(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 R^{i}F(X) is the following: take an injective resolution of X as above, and let K^{i} be the image of the map I^{i1}→I^{i}, which is the same as the kernel of I^{i}→I^{i+1}. Let φ_{i} : I^{i1}→K^{i} be the corresponding surjective map. Then R^{i}F(X) is the cokernel of F(φ_{i}).
Variations
If one starts with a covariant rightexact 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 leftderived functors L_{i}G. For an object X of A we first construct a projective resolution of the form
 <math>\cdots\to P_2\to P_1\to P_0 \to X \to 0<math>
where the P_{i} are projective. We apply G to this sequence, chop off the last term, and compute homology to get L_{i}G(X).
In this case, the long exact sequence will grow "to the left" rather than to the right:
 <math>0\to A \to B \to C \to 0<math>
is turned into
 <math>\cdots\to L_2G(C) \to L_1G(A) \to L_1G(B)\to L_1G(C)\to G(A)\to G(B)\to G(C)\to 0<math>.
Left derived functors are zero on all projective objects.
One may also start with a contravariant leftexact functor F; the resulting rightderived functors are then also contravariant. The short exact sequence
 <math>0\to A \to B \to C \to 0<math>
is turned into the long exact sequence
 <math>0\to F(C)\to F(B)\to F(A)\to R^1F(C) \to R^1F(B) \to R^1F(A)\to R^2F(C)\to \cdots<math>
These right derived functors are zero on projectives and are therefore computed via projective resolutions.
Applications
Sheaf cohomology. If X is a topological space, then the category of all sheaves of abelian groups on X is an abelian category with enough injectives (a result of Grothendieck). The functor which assigns to each such sheaf L the group L(X) of global sections is left exact, and the right derived functors are the sheaf cohomology functors, usually written as H^{ i}(X,L). Slightly more generally: if (X, O_{X}) is a ringed space, then the category of all sheaves of 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.
Ext functors. If R is a ring, then the category of all left Rmodules is an abelian category with enough injectives. If A is a fixed left Rmodule, then the functor Hom(A,) is left exact, and its right derived functors are the Ext functors Ext_{R}^{i}(A,B).
Tor functors. The category of left Rmodules also has enough projectives. If A is a fixed right Rmodule, then the tensor product with A gives a right exact covariant functor on the category of left Rmodules; its left derivatives are the Tor functors Tor^{R}_{i}(A,B).
Group cohomology. Let G be a group. A Gmodule M is an abelian group M together with a group action of G on M as a group of automorphisms. This is the same as a module over the group ring ZG. The Gmodules form an abelian category with enough injectives. We write M^{G} for the subgroup of M consisting of all elements of M that are held fixed by G. This is a leftexact functor, and its right derived functors are the group cohomology functors, typically written as H^{ i}(G,M).
Naturality
Derived functors and the long exact sequences are "natural" in several technical senses.
First, given a commutative diagram of the form
 Missing image
Two_exact_sequences.png
image:two_exact_sequences.png
(where the rows are exact), the two resulting long exact sequences are related by commuting squares:
Missing image
Two_long_exact_sequences.png
image:two_long_exact_sequences.png
Second, suppose η : F → G is a natural transformation from the left exact functor F to the left exact functor G. Then natural transformations R^{i}η : R^{i}F → R^{i}G are induced, and indeed R^{i} becomes a functor from the functor category of all left exact functors from A to B to the full functor category of all functors from A to B. Furthermore, this functor is compatible with the long exact sequences in the following sense: if
is a short exact sequence, then a commutative diagram
is induced.
Both of these naturalities follow form the naturality of the sequence provided by the snake lemma.
Generalization
The more modern (and more general) approach to derived functors uses the language of derived categories.