Derived functor

In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, certain functor

Functor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

s 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

Abelian category

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

 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

Injective object

In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in homotopy theory and in theory of model categories...

, then the sequence splits

Splitting lemma

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for short exact sequence are equivalent....

. 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

Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....

 A → I where I is an injective object

Injective object

In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in homotopy theory and in theory of model categories...

 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

Chain complex

In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...

Note: this is in general not an exact sequence anymore. But we can compute its homology

Homology (mathematics)

In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...

 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

Snake lemma

The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology...

.

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

Projective module

In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...

), 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). As before, L₀G(X) = G(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

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

, then the category of all sheaves

Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...

 of abelian group

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s 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

Sheaf cohomology

In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...

 functors, usually written as H ⁱ(X,L). Slightly more generally: if (X, O_X) is a ringed space

Ringed space

In mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the 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

Étale cohomology

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...

 is another cohomology theory for sheaves over a scheme.

Ext functors. If R is a ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

, then the category of all left R-modules

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

 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 functor

Ext functor

In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics.- Definition and computation :...

s 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

Tensor product

In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

 with A gives a right exact covariant functor on the category of left R-modules; its left derivatives are the Tor functor

Tor functor

In homological algebra, the Tor functors are the derived functors of the tensor product functor. They were first defined in generality to express the Künneth theorem and universal coefficient theorem in algebraic topology....

s Tor^R_i(A,B).

Group cohomology. Let G be a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

. A G-module

G-module

In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G...

 M is an abelian group

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

 M together with a group action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 of G on M as a group of automorphisms. This is the same as a module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

 over the group ring

Group ring

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...

 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

Group cohomology

In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...

 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

Commutative diagram

In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...

 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

Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

 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

Functor category

In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors...

 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 from the naturality of the sequence provided by the snake lemma

Snake lemma

The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology...

.

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

Derived category

In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...

.