In

_{0}, ''A''_{1}, ''A''_{2}, ''A''_{3}, ''A''_{4}, ... connected by homomorphisms (called boundary operators or differentials) , such that the composition of any two consecutive maps is the zero map. Explicitly, the differentials satisfy , or with indices suppressed, . The complex may be written out as follows.
::$\backslash cdots\; \backslash xleftarrow\; A\_0\; \backslash xleftarrow\; A\_1\; \backslash xleftarrow\; A\_2\; \backslash xleftarrow\; A\_3\; \backslash xleftarrow\; A\_4\; \backslash xleftarrow\; \backslash cdots$
The cochain complex $(A^\backslash bullet,\; d^\backslash bullet)$ is the dual (category theory), dual notion to a chain complex. It consists of a sequence of abelian groups or modules ..., ''A''^{0}, ''A''^{1}, ''A''^{2}, ''A''^{3}, ''A''^{4}, ... connected by homomorphisms satisfying . The cochain complex may be written out in a similar fashion to the chain complex.
::$\backslash cdots\; \backslash xrightarrow\; A^0\; \backslash xrightarrow\; A^1\; \backslash xrightarrow\; A^2\; \backslash xrightarrow\; A^3\; \backslash xrightarrow\; A^4\; \backslash xrightarrow\; \backslash cdots$
The index ''n'' in either ''A''_{''n''} or ''A''^{''n''} is referred to as the degree (or dimension). The difference between chain and cochain complexes is that, in chain complexes, the differentials decrease dimension, whereas in cochain complexes they increase dimension. All the concepts and definitions for chain complexes apply to cochain complexes, except that they will follow this different convention for dimension, and often terms will be given the prefix ''co-''. In this article, definitions will be given for chain complexes when the distinction is not required.
A bounded chain complex is one in which almost all#cardinality, almost all the ''A''_{''n''} are 0; that is, a finite complex extended to the left and right by 0. An example is the chain complex defining the simplicial homology of a finite simplicial complex. A chain complex is bounded above if all modules above some fixed degree ''N'' are 0, and is bounded below if all modules below some fixed degree are 0. Clearly, a complex is bounded both above and below if and only if the complex is bounded.
The elements of the individual groups of a (co)chain complex are called (co)chains. The elements in the kernel of ''d'' are called (co)cycles (or closed elements), and the elements in the image of ''d'' are called (co)boundaries (or exact elements). Right from the definition of the differential, all boundaries are cycles. The ''n''-th (co)homology group ''H''_{''n''} (''H''^{''n''}) is the group of (co)cycles modulo (jargon)#structures, modulo (co)boundaries in degree ''n'', that is,
::$H\_n\; =\; \backslash ker\; d\_/\backslash mbox\; d\_\; \backslash quad\; \backslash left(H^n\; =\; \backslash ker\; d^/\backslash mbox\; d^\; \backslash right)$

_{''k''}, ''A''_{''k''+1}, ''A''_{''k''+2} may be nonzero. For example, the following chain complex is a short exact sequence.
:$\backslash cdots\; \backslash xrightarrow\; \backslash ;\; 0\; \backslash ;\; \backslash xrightarrow\; \backslash ;\; \backslash mathbf\; \backslash ;\; \backslash xrightarrow\; \backslash ;\; \backslash mathbf\; \backslash twoheadrightarrow\; \backslash mathbf/p\backslash mathbf\; \backslash ;\; \backslash xrightarrow\; \backslash ;\; 0\; \backslash ;\; \backslash xrightarrow\; \backslash cdots$
In the middle group, the closed elements are the elements pZ; these are clearly the exact elements in this group.

_{*} between the singular homology of ''X'' and ''Y'' as well. When ''X'' and ''Y'' are both equal to the n-sphere, ''n''-sphere, the map induced on homology defines the Degree of a continuous mapping#From Sn to Sn, degree of the map ''f''.
The concept of chain map reduces to the one of boundary through the construction of the Mapping cone (homological algebra), cone of a chain map.

_{''A''} + ''d''_{''B''}''h'' is easily verified to induce the zero map on homology, for any ''h''. It immediately follows that ''f'' and ''g'' induce the same map on homology. One says ''f'' and ''g'' are chain homotopic (or simply homotopic), and this property defines an equivalence relation between chain maps.
Let ''X'' and ''Y'' be topological spaces. In the case of singular homology, a homotopy between continuous maps induces a chain homotopy between the chain maps corresponding to ''f'' and ''g''. This shows that two homotopic maps induce the same map on singular homology. The name "chain homotopy" is motivated by this example.

_{''n''}(''X'') for Natural number, natural ''n'' to be the free abelian group formally generated by singular homology, singular n-simplices in ''X'', and define the boundary map $\backslash partial\_n:\; C\_n(X)\; \backslash to\; C\_(X)$ to be
::$\backslash partial\_n\; :\; \backslash ,\; (\backslash sigma:\; [v\_0,\backslash ldots,v\_n]\; \backslash to\; X)\; \backslash mapsto\; (\backslash sum\_^n\; (-1)^i\; \backslash sigma:\; [v\_0,\backslash ldots,\; \backslash hat\; v\_i,\; \backslash ldots,\; v\_n]\; \backslash to\; X)$
where the hat denotes the omission of a vertex (geometry), vertex. That is, the boundary of a singular simplex is the alternating sum of restrictions to its faces. It can be shown that ∂^{2} = 0, so $(C\_\backslash bullet,\; \backslash partial\_\backslash bullet)$ is a chain complex; the singular homology $H\_\backslash bullet(X)$ is the homology of this complex.
Singular homology is a useful invariant of topological spaces up to homotopy#homotopy equivalence, homotopy equivalence. The degree zero homology group is a free abelian group on the connected space#Path connectedness, path-components of ''X''.

^{''k''}(''M'') under addition.
The exterior derivative ''d'' maps Ω^{''k''}(''M'') to Ω^{''k''+1}(''M''), and ''d'' = 0 follows essentially from symmetry of second derivatives, so the vector spaces of ''k''-forms along with the exterior derivative are a cochain complex.
:$\backslash Omega^0(M)\backslash \; \backslash stackrel\backslash \; \backslash Omega^1(M)\; \backslash to\; \backslash Omega^2(M)\; \backslash to\; \backslash Omega^3(M)\; \backslash to\; \backslash cdots$
The cohomology of this complex is called the de Rham cohomology of ''X''. The homology group in dimension zero is isomorphic to the vector space of locally constant functions from ''M'' to R. Thus for a compact manifold, this is the real vector space whose dimension is the number of connected components of ''M''.
smoothness#smooth functions between manifolds, Smooth maps between manifolds induce chain maps, and smooth homotopies between maps induce chain homotopies.

_{''K''}, where ''K'' is a commutative ring.
If ''V'' = ''V''$\_*$ and ''W'' = ''W''$\_*$ are chain complexes, their tensor product $V\; \backslash otimes\; W$ is a chain complex with degree ''n'' elements given by
:$(V\; \backslash otimes\; W)\_n\; =\; \backslash bigoplus\_\; V\_i\; \backslash otimes\; W\_j$
and differential given by
: $\backslash partial\; (a\; \backslash otimes\; b)\; =\; \backslash partial\; a\; \backslash otimes\; b\; +\; (-1)^\; a\; \backslash otimes\; \backslash partial\; b$
where ''a'' and ''b'' are any two homogeneous vectors in ''V'' and ''W'' respectively, and $\backslash left,\; a\backslash $ denotes the degree of ''a''.
This tensor product makes the category Ch_{''K''} into a symmetric monoidal category. The identity object with respect to this monoidal product is the base ring ''K'' viewed as a chain complex in degree 0. The braided monoidal category, braiding is given on simple tensors of homogeneous elements by
:$a\; \backslash otimes\; b\; \backslash mapsto\; (-1)^\; b\; \backslash otimes\; a$
The sign is necessary for the braiding to be a chain map.
Moreover, the category of chain complexes of ''K''-modules also has closed monoidal category, internal Hom: given chain complexes ''V'' and ''W'', the internal Hom of ''V'' and ''W'', denoted Hom(''V'',''W''), is the chain complex with degree ''n'' elements given by $\backslash Pi\_\backslash text\_K\; (V\_i,W\_)$ and differential given by
: $(\backslash partial\; f)(v)\; =\; \backslash partial(f(v))\; -\; (-1)^\; f(\backslash partial(v))$.
We have a natural isomorphism
:$\backslash text(A\backslash otimes\; B,\; C)\; \backslash cong\; \backslash text(A,\backslash text(B,C))$

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 general consensus abo ...

, a chain complex is an 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 ...

that consists of a sequence of abelian group
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 g ...

s (or 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 a sequence of between consecutive groups such that the image
An SAR radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teide volcano. The city of Santa Cruz de Tenerife is visible as the purple and white area on the lower right edge of the island. Lava flows ...

of each homomorphism is included in 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. Associated to a chain complex is its Homology (mathematics), homology, which describes how the images are included in the kernels.
A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology.
In algebraic topology, the singular chain complex of a topological space X is constructed using continuous function#continuous functions between topological spaces, continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex. The homology of this chain complex is called the singular homology of X, and is a commonly used topological invariant, invariant of a topological space.
Chain complexes are studied in homological algebra, but are used in several areas of mathematics, including abstract algebra, Galois theory, differential geometry and algebraic geometry. They can be defined more generally in abelian categories.
Definitions

A chain complex $(A\_\backslash bullet,\; d\_\backslash bullet)$ is a sequence of abelian groups or modules ..., ''A''Exact sequences

An exact sequence (or exact complex) is a chain complex whose homology groups are all zero. This means all closed elements in the complex are exact. A short exact sequence is a bounded exact sequence in which only the groups ''A''Chain maps

A chain map ''f'' between two chain complexes $(A\_\backslash bullet,\; d\_)$ and $(B\_\backslash bullet,\; d\_)$ is a sequence $f\_\backslash bullet$ of homomorphisms $f\_n\; :\; A\_n\; \backslash rightarrow\; B\_n$ for each ''n'' that commutes with the boundary operators on the two chain complexes, so $d\_\; \backslash circ\; f\_n\; =\; f\_\; \backslash circ\; d\_$. This is written out in the following commutative diagram. : A chain map sends cycles to cycles and boundaries to boundaries, and thus induces a map on homology $(f\_\backslash bullet)\_*:H\_\backslash bullet(A\_\backslash bullet,\; d\_)\; \backslash rightarrow\; H\_\backslash bullet(B\_\backslash bullet,\; d\_)$. A continuous map ''f'' between topological spaces ''X'' and ''Y'' induces a chain map between the singular chain complexes of ''X'' and ''Y'', and hence induces a map ''f''Chain homotopy

A chain homotopy offers a way to relate two chain maps that induce the same map on homology groups, even though the maps may be different. Given two chain complexes ''A'' and ''B'', and two chain maps , a chain homotopy is a sequence of homomorphisms such that . The maps may be written out in a diagram as follows, but this diagram is not commutative. : The map ''hd''Examples

Singular homology

Let ''X'' be a topological space. Define ''C''de Rham cohomology

The differential form, differential ''k''-forms on any smooth manifold ''M'' form a real number, real vector space called ΩCategory of chain complexes

Chain complexes of ''K''-modules with chain maps form a category (mathematics), category ChFurther examples

*Amitsur complex *A complex used to define Bloch's higher Chow groups *Buchsbaum–Rim complex *Čech complex *Cousin complex *Eagon–Northcott complex *Gersten complex *Graph complexhttps://ncatlab.org/nlab/show/graph+complex *Koszul complex *Moore complex *Schur complexSee also

* Differential graded algebra * Differential graded Lie algebra * Dold–Kan correspondence says there is an equivalence between the category of chain complexes and the category of simplicial abelian groups. * Buchsbaum–Eisenbud acyclicity criterion * Differential graded moduleReferences

* * {{cite book , last=Hatcher , first=Allen , author-link=Allen Hatcher , date=2002 , title=Algebraic Topology , url=https://www.math.cornell.edu/~hatcher/AT/ATpage.html , location=Cambridge , publisher=Cambridge University Press , isbn=0-521-79540-0 Homological algebra Differential topology