In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... 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 nice properties. ... Group theory is that branch of mathematics concerned with the study of groups. ... This is a page about mathematics. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In common usage, an image (from Latin imago) or picture is an artifact that reproduces the likeness of some subject—usually a physical object or a person. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ...

Definition

To be precise, fix an Abelian category (such as the category of Abelian groups or the category of vector spaces over a given field) or some other category with kernels and cokernels (such as the category of all groups). Choose an index set of consecutive integers. Then for each integer i in the index set, let A_i be an object in the category and let f_i be a morphism from A_i to A_i+1. This defines a sequence of objects and morphisms. 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 nice properties. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ... In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, an index set is another name for a function domain. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

The sequence is exact at A_i if the image of f_i−1 is equal to the kernel of f_i: Given a category C and a morphism in C, the image of f is a monomorphism satisfying the following universal property: There exists a morphism such that f = hg. ... In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra. ...

im f_i−1 = ker f_i.

The sequence itself is exact if it is exact at each object (except possibly at the very first and the very last object, where exactness doesn't make sense).

Example

Consider the following sequence of abelian groups: In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...

where 0 denotes the trivial abelian group with a single element, the map from Z to Z is multiplication by 2, and the map from Z to the factor group Z/2Z is given by reducing integers modulo 2. This is indeed an exact sequence: 2 (two) is the natural number following 1 and preceding 3. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

• the image of the map 0→Z is {0}, and the kernel of multiplication by 2 is also {0}, so the sequence is exact at the first Z.
• the image of multiplication by 2 is 2Z, and the kernel of reducing modulo 2 is also 2Z, so the sequence is exact at the second Z.
• the image of reducing modulo 2 is all of Z/2Z, and the kernel of the zero map is also all of Z/2Z, so the sequence is exact at the position Z/2Z

Special cases

To make sense of the definition, it is helpful to consider what it means in relatively simple cases where the sequence is finite and begins or ends with 0.

• The sequence 0 → A → B is exact at A if and only if the map from A to B has kernel {0}, i.e. if and only if that map is a monomorphism.
• Dually, the sequence B → C → 0 is exact at C if and only if the image of the map from B to C is all of C, i.e. if and only if that map is an epimorphism.
• A consequence of these last two facts is that the sequence 0 → X → Y → 0 is exact if and only if the map from X to Y is an isomorphism.

When dealing with exact sequences of groups, it is common to write 1 instead of 0 for the trivial group with a single element. In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In the context of abstract algebra or universal algebra, an epimorphism is simply a homomorphism onto or surjective homomorphism. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. ...

Important are short exact sequences, which are exact sequences of the form

By the above, we know that for any such short exact sequence, f is a monomorphism and g is an epimorphism. Furthermore, the image of f is equal to the kernel of g. It is helpful to think of A as a subobject of B with f being the embedding of A into B, and of C as the corresponding factor object B/A, with the map g being the natural projection from B to B/A (whose kernel is exactly A).

Facts

The splitting lemma states that if the above short exact sequence admits a morphism t: B → A such that t o f is the identity on A or a morphism u: C → B such that g o u is the identity on C, then B is a twisted direct sum of A and C. (For groups, a twisted direct sum is a semidirect product; in an Abelian category, every twisted direct sum is an ordinary direct sum.) In this case, we say that the short exact sequence splits. In mathematics, and more specifically in homological algebra, the splitting lemma states that the following statements regarding the below short exact sequence in any abelian category (with similar results if the objects are non-abelian groups) are equivalent: there exists a map t: B → A such that tq is the... OR logic gate Logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true. ... In abstract algebra, a semidirect product describes a particular way in which a group can be put together from two subgroups. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

The snake lemma shows how a commutative diagram with two exact rows gives rise to a longer exact sequence. The nine lemma is a special case. In mathematics, particularly homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the long exact sequences that are ubiquitous in homological algebra and its applications, for instance in algebraic topology. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ... In mathematics, the nine lemma is a statement about commutative diagrams and exact sequences valid in any abelian category, as well as in the category of groups. ...

The five lemma gives conditions under which the middle map in a commutative diagram with exact rows of length 5 is an isomorphism; the short five lemma is a special case thereof applying to short exact sequences. In mathematics, especially homological algebra and other applications of Abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. ... In mathematics, especially homological algebra and other applications of Abelian category theory, the short five lemma is a special case of the five lemma. ...

The importance of short exact sequences is underlined by the fact that every exact sequence results from "weaving together" several overlapping short exact sequences. Consider for instance the exact sequence

and define

Then we obtain a commutative diagram in which all the diagonals are short exact sequences:

Conversely, given any list of overlapping short exact sequences, their middle terms form an exact sequence in the same manner.

Applications of exact sequences

In the theory of abelian categories, short exact sequences are often used as a convenient language to talk about sub- and factor objects.

The extension problem is essentially the question, given the end terms A and C of a short exact sequence, what possibilities exist for the middle term B? In the category of abelian groups, this is equivalent to the question, what groups B have A as a normal subgroup and C as the corresponding factor group? This problem is important in the classification of groups. In group theory, if the factor group G/K is isomorphic to H, one says that G is an extension of H by K. To consider some examples, if G = H × K, then G is an extension of both H and K. More generally, if G is a semidirect product... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...

Notice that in an exact sequence, the composition f_i+1 o f_i maps A_i to 0 in A_i+2, so every exact sequence is a chain complex. Furthermore, only f_i-images of elements of A_i are mapped to 0 by f_i+1, so the homology of this chain complex is trivial. More succinctly: In homological algebra, a chain complex is a sequence of abelian groups or modules A0, A1, A2. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...

Exact sequences are precisely those chain complexes which are acyclic.

Given any chain complex, its homology can therefore be thought of as a measure of the degree to which it fails to be exact. Acyclic can mean any of the following: In chemistry, an acyclic compound is a hydrocarbon compound having an open chain. ...

If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a long exact sequence (i.e. an exact sequence indexed by the natural numbers) by repeated application of the snake lemma. This is explained in the article on homology. It comes up in algebraic topology in the study of relative homology; the Mayer-Vietoris sequence is another example. Long exact sequences induced by short exact sequences are also characteristic of derived functors. In mathematics, particularly homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the long exact sequences that are ubiquitous in homological algebra and its applications, for instance in algebraic topology. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology which is somewhat analogous to forming quotient objects in other categories. ... In algebraic topology and related branches of mathematics, the Mayer-Vietoris sequence (named after Walther Mayer and Leopold Vietoris) is an exact sequence that often helps one to compute homology groups. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ...

Exact functors are functors that transform exact sequences into exact sequences. In homological algebra, an exact functor is one which preserves exact sequences. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...