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 is not related in any way to the derivatives of calculus.)

Table of contents

1 Motivation 2 Construction and first properties 3 Variations 4 Applications 5 Naturality 6 Generalization

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

is a short exact sequence in A, then applying F yields the exact sequence

and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, 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 RⁱF: A → B, and the above sequence continues like so:

From this we see that F is an exact functor if and only if R¹F = 0; so in a sense the right exact functors 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 R¹F(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 left-exact 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

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 complex

Note: this is in general not an exact sequence anymore. But we can compute its homology at the i-th spot (kernel of the (i+1)-th map modulo image of the i-th map), and the result we call 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.

The above-mentioned 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 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.

Variations

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 L_iG. For an object X of A we first construct a projective resolution of the form

where the P_i are projective. We apply G to this sequence, chop off the last term, and compute homology to get L_iG(X).

In this case, the long exact sequence will grow "to the left" rather than to the right:

is turned into

.

Left derived functors are zero on all projective objects.

One may also start with a contravariant left-exact functor F; the resulting right-derived functors are then also contravariant. The short exact sequence

is turned into the long exact sequence

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 R-modules is an abelian category with enough injectives. If A is a fixed left R-module, then the functor Hom(A,-) is left exact, and its right derived functors are the Ext functors Ext_Rⁱ(A,B).

Tor functors. The category of left R-modules also has enough projectives. If A is a fixed right R-module, then the tensor product with A gives a right exact covariant functor on the category of left R-modules; its left derivatives are the Tor functors Tor^R_i(A,B).

Group cohomology. Let G be a group. A G-module 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 G-modules 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 left-exact 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

image:two_exact_sequences.png

(where the rows are exact), the two resulting long exact sequences are related by commuting squares:

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 Rⁱη : RⁱF → RⁱG are induced, and indeed Rⁱ 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

image:two_long_exact_sequences2.png

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.