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 sequeence

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 natural number i, there is a functor RⁱF: A → B, and the above sequence continues like so:

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 enouch 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

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

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. 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.

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 sheave 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 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 Extⁱ(A,B).

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

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

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

is induced.

Generalization

The more modern (and more general) approach to derived functors uses the language of derived categories.