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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.

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 0 → A → B → C → 0 is a short exact sequence in A, then applying F yields the exact sequence 0 → F(A) → F(B) → F(C) 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 canonical 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: 0 → F(A) → F(B) → F(C) → R¹F(A) → R¹F(B) → R¹F(C) → R²F(A) → R²F(B) → ... . From this we see that F is an exact functor if and only if R¹F = 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 any additive functor to a split sequence results in a split sequence, so in particular 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 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

$0\to X\to I^0\to I^1\to I^2\to \cdots$

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

$0\to F(I^0)\to F(I^1)\to F(I^2)\to \cdots$

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 I^i-1→Iⁱ (for i=0, define I^i-1=0), which is the same as the kernel of Iⁱ→Iⁱ⁺¹. Let φ_i : I^i-1→Kⁱ be the corresponding surjective map. Then RⁱF(X) is the cokernel of F(φ_i).

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

$\cdots \to P_2\to P_1\to P_0\to X\to 0$

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). As before, L₀G(X) = G(X).

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

$0\to A\to B\to C\to 0$

is turned into

$\cdots \to L_{2}G(C)\to L_{1}G(A)\to L_{1}G(B)\to L_{1}G(C)\to G(A)\to G(B)\to G(C)\to 0$.

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

$0\to A\to B\to C\to 0$

is turned into the long exact sequence

$0\to F(C)\to F(B)\to F(A)\to R^{1}F(C)\to R^{1}F(B)\to R^{1}F(A)\to R^{2}F(C)\to \cdots$

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. 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.

Étale cohomology is another cohomology theory for sheaves over a scheme.

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,-).

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,-).

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

$\begin{array}{ccccccccc}0& \stackrel{}{\to }& A_1& \stackrel{f_1}{\to }& B_1& \stackrel{g_1}{\to }& C_1& \stackrel{}{\to }& 0\\ & & \alpha \downarrow & & \beta \downarrow & & \gamma \downarrow & & \\ 0& \stackrel{}{\to }& A_2& \stackrel{f_2}{\to }& B_2& \stackrel{g_2}{\to }& C_2& \stackrel{}{\to }& 0\end{array}$

(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

$0\stackrel{}{\to }A\stackrel{f}{\to }B\stackrel{g}{\to }C\stackrel{}{\to }0$

is a short exact sequence, then a commutative diagram

is induced.

Both of these naturalities follow from the naturality of the sequence provided by the snake lemma.

Conversely, the following characterization of derived functors holds: given a family of functors Rⁱ: A → B, satisfying the above, i.e. mapping short exact sequences to long exact sequences, such that for every injective object I of A, Rⁱ(I)=0 for every positive i, then these functors are the right derived functors of R⁰.

Generalization

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

References

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