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An exact sequence is a sequence of
morphisms In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
between objects (for example,
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
, rings,
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
, and, more generally, objects of an
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 prototypical example of an abelian category is the category of ...
) such that the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of one morphism equals the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
of the next.


Definition

In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
s is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i, i.e., if the image of each homomorphism is equal to the kernel of the next. The sequence of groups and homomorphisms may be either finite or infinite. A similar definition can be made for other
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
s. For example, one could have an exact sequence of
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s and
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
s, or of modules and
module homomorphism In algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if ''M'' and ''N'' are left modules over a ring ''R'', then a function f: M \to N is called an ''R''-''module homomorphism'' or an '' ...
s. More generally, the notion of an exact sequence makes sense in any
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
with
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...
s and
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...
s, and more specially in
abelian categories 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 prototypical example of an abelian category is the category of abe ...
, where it is widely used.


Simple cases

To understand the definition, it is helpful to consider relatively simple cases where the sequence is of group homomorphisms, is finite, and begins or ends with the
trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
. Traditionally, this, along with the single identity element, is denoted 0 (additive notation, usually when the groups are abelian), or denoted 1 (multiplicative notation). * Consider the sequence 0 → ''A'' → ''B''. The image of the leftmost map is 0. Therefore the sequence is exact if and only if the rightmost map (from ''A'' to ''B'') has kernel ; that is, if and only if that map is a
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...
(injective, or one-to-one). * Consider the dual sequence ''B'' → ''C'' → 0. The kernel of the rightmost map is ''C''. Therefore the sequence is exact if and only if the image of the leftmost map (from ''B'' to ''C'') is all of ''C''; that is, if and only if that map is an
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f ...
(surjective, or onto). * Therefore, the sequence 0 → ''X'' → ''Y'' → 0 is exact if and only if the map from ''X'' to ''Y'' is both a monomorphism and epimorphism (that is, a
bimorphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
), and so usually an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
from ''X'' to ''Y'' (this always holds in
exact categories In mathematics, an exact category is a concept of category theory due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and ...
like Set).


Short exact sequence

Short exact sequences are exact sequences of the form :0 \to A \xrightarrow B \xrightarrow C \to 0. As established above, 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 In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory ...
of ''B'' with ''f'' embedding ''A'' into ''B'', and of ''C'' as the corresponding factor object (or
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
), ''B''/''A'', with ''g'' inducing an isomorphism :C \cong B/\operatorname(f) = B/\operatorname(g) The short exact sequence :0 \to A \xrightarrow B \xrightarrow C \to 0\, is called
split Split(s) or The Split may refer to: Places * Split, Croatia, the largest coastal city in Croatia * Split Island, Canada, an island in the Hudson Bay * Split Island, Falkland Islands * Split Island, Fiji, better known as Hạfliua Arts, entertai ...
if there exists a homomorphism ''h'' : ''C'' → ''B'' such that the composition ''g'' ∘ ''h'' is the identity map on ''C''. It follows that if these are
abelian group In mathematics, 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 the order in which they are written. That is, the group operation is comm ...
s, ''B'' is isomorphic to the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of ''A'' and ''C'': :B \cong A \oplus C.


Long exact sequence

A general exact sequence is sometimes called a long exact sequence, to distinguish from the special case of a short exact sequence. A long exact sequence is equivalent to a family of short exact sequences in the following sense: Given a long sequence with ''n ≥'' 2, we can split it up into the short sequences where K_i = \operatorname(f_i) for every i. By construction, the sequences ''(2)'' are exact at the K_i's (regardless of the exactness of ''(1)''). Furthermore, ''(1)'' is a long exact sequence if and only if ''(2)'' are all short exact sequences.


Examples


Integers modulo two

Consider the following sequence of abelian groups: :\mathbf \mathrel \mathbf \twoheadrightarrow \mathbf/2\mathbf The first homomorphism maps each element ''i'' in the set of integers Z to the element 2''i'' in Z. The second homomorphism maps each element ''i'' in Z to an element ''j'' in the quotient group; that is, . Here the hook arrow \hookrightarrow indicates that the map 2× from Z to Z is a monomorphism, and the two-headed arrow \twoheadrightarrow indicates an epimorphism (the map mod 2). This is an exact sequence because the image 2Z of the monomorphism is the kernel of the epimorphism. Essentially "the same" sequence can also be written as :2\mathbf \mathrel \mathbf \twoheadrightarrow \mathbf/2\mathbf In this case the monomorphism is 2''n'' ↦ 2''n'' and although it looks like an identity function, it is not onto (that is, not an epimorphism) because the odd numbers don't belong to 2Z. The image of 2Z through this monomorphism is however exactly the same subset of Z as the image of Z through ''n'' ↦ 2''n'' used in the previous sequence. This latter sequence does differ in the concrete nature of its first object from the previous one as 2Z is not the same set as Z even though the two are isomorphic as groups. The first sequence may also be written without using special symbols for monomorphism and epimorphism: :0 \to \mathbf \mathrel \mathbf \longrightarrow \mathbf/2\mathbf \to 0 Here 0 denotes the trivial group, the map from Z to Z is multiplication by 2, and the map from Z to the
factor group Factor, a Latin word meaning "who/which acts", may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, s ...
Z/2Z is given by reducing integers
modulo In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation). Given two positive numbers and , modulo (often abbreviated as ) is ...
2. This is indeed an exact sequence: * the image of the map 0 → Z is , and the kernel of multiplication by 2 is also , 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 Z/2Z, and the kernel of the zero map is also Z/2Z, so the sequence is exact at the position Z/2Z. The first and third sequences are somewhat of a special case owing to the infinite nature of Z. It is not possible for a
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or ma ...
to be mapped by inclusion (that is, by a monomorphism) as a proper subgroup of itself. Instead the sequence that emerges from the first isomorphism theorem is :1 \to N \to G \to G/N \to 1 As a more concrete example of an exact sequence on finite groups: :1 \to C_n \to D_ \to C_2 \to 1 where C_n is the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order ''n'' and D_ is the
dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
of order 2''n'', which is a non-abelian group.


Intersection and sum of modules

Let and be two
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
s of a ring . Then :0 \to I\cap J \to I\oplus J \to I + J \to 0 is an exact sequence of -modules, where the module homomorphism I\cap J \to I\oplus J maps each element of I\cap J to the element of the
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
I\oplus J, and the homomorphism I\oplus J \to I+J maps each element of I\oplus J to . These homomorphisms are restrictions of similarly defined homomorphisms that form the short exact sequence :0\to R \to R\oplus R \to R \to 0 Passing to quotient modules yield another exact sequence :0\to R/(I\cap J) \to R/I \oplus R/J \to R/(I+J) \to 0


Grad, curl and div in differential geometry

Another example can be derived from
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
, especially relevant for work on the
Maxwell equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. Th ...
. Consider the
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
L^2 of scalar-valued square-integrable functions on three dimensions \left\lbrace f:\mathbb^3 \to \mathbb \right\rbrace. Taking the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of a function f\in\mathbb_1 moves us to a subset of \mathbb_3, the space of vector valued, still square-integrable functions on the same domain \left\lbrace f:\mathbb^3\to\mathbb^3 \right\rbrace — specifically, the set of such functions that represent conservative vector fields. (The generalized
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
has preserved integrability.) First, note the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was ...
of all such fields is zero — since :\operatorname (\operatorname f ) \equiv \nabla \times (\nabla f) = 0 for all such . However, this only proves that the image of the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
is a subset of the kernel of the curl. To prove that they are in fact the same set, prove the converse: that if the curl of a vector field \vec is 0, then \vec is the gradient of some scalar function. This follows almost immediately from
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
(see the proof at
conservative force In physics, a conservative force is a force with the property that the total work done in moving a particle between two points is independent of the path taken. Equivalently, if a particle travels in a closed loop, the total work done (the sum ...
.) The image of the
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
is then precisely the kernel of the curl, and so we can then take the curl to be our next morphism, taking us again to a (different) subset of \mathbb_3. Similarly, we note that :\operatorname \left(\operatorname \vec\right) \equiv \nabla \cdot \nabla \times \vec = 0, so the image of the curl is a subset of the kernel of the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
. The converse is somewhat involved: Having thus proved that the image of the curl is precisely the kernel of the divergence, this morphism in turn takes us back to the space we started from L^2. Since definitionally we have landed on a space of integrable functions, any such function can (at least formally) be integrated in order to produce a vector field which divergence is that function — so the image of the divergence is the entirety of L^2, and we can complete our sequence: :0 \to L^2 \mathrel \mathbb_3 \mathrel \mathbb_3 \mathrel L^2 \to 0 Equivalently, we could have reasoned in reverse: in a
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
space, a curl-free vector field (a field in the kernel of the curl) can always be written as a gradient of a scalar function (and thus is in the image of the gradient). Similarly, a
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
less field can be written as a curl of another field. (Reasoning in this direction thus makes use of the fact that 3-dimensional space is topologically trivial.) This short exact sequence also permits a much shorter proof of the validity of the
Helmholtz decomposition In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into ...
that does not rely on brute-force vector calculus. Consider the subsequence :0 \to L^2 \mathrel \mathbb_3 \mathrel \operatorname(\operatorname) \to 0. Since the divergence of the gradient is the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
, and since the Hilbert space of square-integrable functions can be spanned by the eigenfunctions of the Laplacian, we already see that some inverse mapping \nabla^:\mathbb_3 \to L^2 must exist. To explicitly construct such an inverse, we can start from the definition of the vector Laplacian :\nabla^2 \vec = \nabla\left(\nabla\cdot\vec\right) + \nabla\times\left(\nabla\times\vec\right) Since we are trying to construct an identity mapping by composing some function with the gradient, we know that in our case \nabla\times\vec = \operatorname\left(\vec\right) = 0. Then if we take the divergence of both sides :\begin \nabla\cdot\nabla^2\vec & = \nabla\cdot\nabla\left(\nabla\cdot\vec\right) \\ & = \nabla^2\left(\nabla\cdot\vec\right) \\ \end we see that if a function is an eigenfunction of the vector Laplacian, its divergence must be an eigenfunction of the scalar Laplacian with the same eigenvalue. Then we can build our inverse function \nabla^ simply by breaking any function in \mathbb_3 into the vector-Laplacian eigenbasis, scaling each by the inverse of their eigenvalue, and taking the divergence; the action of \nabla^\circ\nabla is thus clearly the identity. Thus by the splitting lemma, :\mathbb_3 \cong L^2 \oplus \operatorname(\operatorname), or equivalently, any square-integrable vector field on \mathbb^3 can be broken into the sum of a gradient and a curl — which is what we set out to prove.


Properties

The splitting lemma states that if the short exact sequence :0 \to A \;\xrightarrow\; B \;\xrightarrow\; C \to 0 admits a morphism such that is the identity on or a morphism such that is the identity on , then is a
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mor ...
of and (for non-commutative groups, this is a
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in wh ...
). One says that such a short exact sequence ''splits''. The
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 ...
shows how a
commutative diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
with two exact rows gives rise to a longer exact sequence. The nine lemma is a special case. 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 In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma. It states that for the following commutative diagram (in any abelian category, or in the c ...
is a special case thereof applying to short exact sequences. 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 :A_1\to A_2\to A_3\to A_4\to A_5\to A_6 which implies that there exist objects ''Ck'' in the category such that :C_k \cong \ker (A_k\to A_) \cong \operatorname (A_\to A_k). Suppose in addition that the cokernel of each morphism exists, and is isomorphic to the image of the next morphism in the sequence: :C_k \cong \operatorname (A_\to A_) (This is true for a number of interesting categories, including any abelian category such as the abelian groups; but it is not true for all categories that allow exact sequences, and in particular is not true for the
category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories There a ...
, in which coker(''f'') : ''G'' → ''H'' is not ''H''/im(''f'') but H / ^H, the quotient of ''H'' by the
conjugate closure In group theory, the normal closure of a subset S of a group G is the smallest normal subgroup of G containing S. Properties and description Formally, if G is a group and S is a subset of G, the normal closure \operatorname_G(S) of S is the ...
of im(''f'').) Then we obtain a commutative diagram in which all the diagonals are short exact sequences: : The only portion of this diagram that depends on the cokernel condition is the object C_7 and the final pair of morphisms A_6 \to C_7\to 0. If there exists any object A_ and morphism A_k \to A_ such that A_ \to A_k \to A_ is exact, then the exactness of 0 \to C_k \to A_k \to C_ \to 0 is ensured. Again taking the example of the category of groups, the fact that im(''f'') is the kernel of some homomorphism on ''H'' implies that it is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
, which coincides with its conjugate closure; thus coker(''f'') is isomorphic to the image ''H''/im(''f'') of the next morphism. 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 In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\overs ...
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 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. See also
Outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a ...
. Notice that in an exact sequence, the composition ''f''''i''+1 ∘ ''f''''i'' maps ''A''''i'' to 0 in ''A''''i''+2, so every exact sequence is a
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
. 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: :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. 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 (that is, an exact sequence indexed by the natural numbers) on homology by application of the zig-zag lemma. It comes up in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
in the study of
relative homology In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intui ...
; the
Mayer–Vietoris sequence In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due ...
is another example. Long exact sequences induced by short exact sequences are also characteristic of
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. Motivation It was noted in vari ...
s.
Exact functor In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Mu ...
s are
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
s that transform exact sequences into exact sequences.


References

;Citations ;Sources * * {{Topology Homological algebra Additive categories