TheInfoList

An exact sequence is a sequence of morphisms between objects (for example,
groups A group is a number of people 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 identi ...
,
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
,
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 syst ...
, and, more generally, objects of an
abelian category In mathematics, an abelian category is a Category (mathematics), category in which morphisms and Object (category theory), objects can be added and in which Kernel (category theory), kernels and cokernels exist and have desirable properties. The mo ...
) such that the
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of one morphism equals the
kernel Kernel may refer to: Computing * Kernel (operating system) In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s is said to be exact at $G_i$ if $\operatorname\left(f_i\right)=\ker\left(f_\right)$. The sequence is called exact if it is exact at each $G_i$ for all
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. For example, one could have an exact sequence of
vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s and
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s, or of modules and
module homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
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 Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
with
kernel Kernel may refer to: Computing * Kernel (operating system) In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...
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 (mathematics), category in which morphisms and Object (category theory), objects can be added and in which Kernel (category theory), kernels and cokernels exist and have desirable properties. The mot ...
, where it is widely used.

Simple cases

To understand the definition, it is helpful to consider relatively simple cases where the sequence is finite and begins or ends with the
trivial groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. 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 220px In the context of abstract algebra or universal algebra, a monomorphism is an Injective function, injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of catego ...
(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 220px In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ...
(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 (mathematics), map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set t ...
), and thus, in many cases, an
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
from ''X'' to ''Y''.

Short exact sequence

Important are short exact sequences, which 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 Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), sp ...
of ''B'' with ''f'' embedding ''A'' into ''B'', and of ''C'' as the corresponding factor object (or
quotient In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...
), ''B''/''A'', with ''g'' inducing an isomorphism :$C \cong B/\operatorname\left(f\right)$ 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, entertain ...

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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s, ''B'' is isomorphic to the
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
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 $K_i = \operatorname\left(f_i\right)$ 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 A quotient group or factor group is a math Mathematics (from Greek: ) includes the study of such topics as quantity ( number theory), structure (algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit= ...
Z/2Z is given by reducing integers modulo 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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
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 In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for ...

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 The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

of order ''n'' and $D_$ is the
dihedral group In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
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) An ideal is a principle A principle is a proposition or value that is a guide for behavior or evaluation. In law Law is a system A system is a group of Interaction, interacting ...
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 from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
$I\oplus J$, and the homomorphsim $I\oplus J \to I+J$ 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 moduleIn algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...
s yield another exact sequence :$0\to R/\left(I\cap J\right) \to R/I \oplus R/J \to R/\left(I+J\right) \to 0$

Grad, curl and div in differential geometry

Another example can be derived from
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
, especially relevant for work on the
Maxwell equations Maxwell's equations are a set of coupled partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...
. Consider the
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
$L^2$ of scalar-valued square-integrable functions on three dimensions $\left\lbrace f:\mathbb^3 \to \mathbb \right\rbrace$. Taking the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

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' theorem, also known as Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Ba ...
has preserved integrability.) First, note the
curl Curl or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the p ...
of all such fields is zero — since :$\operatorname \left(\operatorname f \right) \equiv \nabla \times \left(\nabla f\right) = 0$ for all such . However, this only proves that the image of the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

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' theorem, also known as Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Ba ...
(see the proof at
conservative force A conservative force is a force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first l ...
.) The image of the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

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 :$\operatorname \left\left(\operatorname \vec\right\right) \equiv \nabla \cdot \nabla \times \vec = 0,$ so the image of the curl is a subset of the kernel of the
divergence In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Produ ...

. 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 s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
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 Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Produ ...

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 Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...
that does not rely on brute-force vector calculus. Consider the subsequence :$0 \to L^2 \mathrel \mathbb_3 \mathrel \operatorname\left(\operatorname\right) \to 0.$ Since the divergence of the gradient is the
Laplacian In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, 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 :$\nabla^2 \vec = \nabla\left\left(\nabla\cdot\vec\right\right) + \nabla\times\left\left(\nabla\times\vec\right\right)$ Since we are trying to construct an identity mapping by composing some function with the gradient, we know that in our case $\nabla\times\vec = \operatorname\left\left(\vec\right\right) = 0$. Then if we take the divergence of both sides :$\begin \nabla\cdot\nabla^2\vec & = \nabla\cdot\nabla\left\left(\nabla\cdot\vec\right\right) \\ & = \nabla^2\left\left(\nabla\cdot\vec\right\right) \\ \end$ 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 In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are logical equivalence, equivalent for a short exact sequence 0 \longrightarrow A \overset B \overset ...
, :$\mathbb_3 \cong L^2 \oplus \operatorname\left(\operatorname\right)$, 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 In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are logical equivalence, equivalent for a short exact sequence 0 \longrightarrow A \overset B \overset ...
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 from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of and (for non-commutative groups, this is a
semidirect product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
). One says that such a short exact sequence ''splits''. The
snake lemma The snake lemma is a tool used in mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their cha ...
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 (category theory), diagram such that all directed paths in the diagram with the same start an ...

with two exact rows gives rise to a longer exact sequence. The 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 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 :$C_k \cong \ker \left(A_k\to A_\right) \cong \operatorname \left(A_\to A_k\right)$. Suppose in addition that the cokernel of each morphism exists, and is isomorphic to the image of the next morphism in the sequence: :$C_k \cong \operatorname \left(A_\to A_\right)$ (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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...
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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
, 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 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 groupIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. 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 Acyclic may refer to: * In chemistry, a compound which is an open-chain compound, e.g. alkanes and acyclic aliphatic compounds * In mathematics: ** A graph without a Cycle (graph theory), cycle, especially *** A directed acyclic graph ** An 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 lemmaIn mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category. Statement In an abelia ...
. It comes up in
algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
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 topological pair, pairs of spaces. The relative homology is useful and important in sev ...
; the Mayer–Vietoris sequence is another example. Long exact sequences induced by short exact sequences are also characteristic of
derived functorIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s.
Exact functor In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s are
functor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s that transform exact sequences into exact sequences.

References

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