Inmathematics, achain complex is analgebraic structure that consists of a sequence ofabelian groups (ormodules) and a sequence ofhomomorphisms between consecutive groups such that theimage of each homomorphism is contained in thekernel of the next. Associated to a chain complex is itshomology, which is (loosely speaking) a measure of the failure of a chain complex to beexact.

Acochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called itscohomology.

Inalgebraic topology, the singular chain complex of atopological space X is constructed usingcontinuous maps from asimplex 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 thesingular homology of X, and is a commonly usedinvariant of a topological space.

Chain complexes are studied inhomological algebra, but are used in several areas of mathematics, includingabstract algebra,Galois theory,differential geometry andalgebraic geometry. They can be defined more generally inabelian categories.

Achain complex${\displaystyle (A_{\bullet },d_{\bullet })}$ is a sequence of abelian groups or modules ...,A₀,A₁,A₂,A₃,A₄, ... connected by homomorphisms (calledboundary operators ordifferentials)d_n :A_n →A_n−1, such that the composition of any two consecutive maps is the zero map. Explicitly, the differentials satisfyd_n ∘d_n+1 = 0, or with indices suppressed,d² = 0. The complex may be written out as follows.

${\displaystyle \cdots \stackrel{d_0}{\leftarrow }{A}_0\stackrel{d_1}{\leftarrow }{A}_1\stackrel{d_2}{\leftarrow }{A}_2\stackrel{d_3}{\leftarrow }{A}_3\stackrel{d_4}{\leftarrow }{A}_4\stackrel{d_5}{\leftarrow }\cdots }$

Thecochain complex${\displaystyle (A^{\bullet },d^{\bullet })}$ is thedual notion to a chain complex. It consists of a sequence of abelian groups or modules ...,A⁰,A¹,A²,A³,A⁴, ... connected by homomorphismsdⁿ :Aⁿ →Aⁿ⁺¹ satisfyingdⁿ⁺¹ ∘dⁿ = 0. The cochain complex may be written out in a similar fashion to the chain complex.

${\displaystyle \cdots \stackrel{d^{-1}}{\to }{A}^0\stackrel{d^0}{\to }{A}^1\stackrel{d^1}{\to }{A}^2\stackrel{d^2}{\to }{A}^3\stackrel{d^3}{\to }{A}^4\stackrel{d^4}{\to }\cdots }$

The indexn in eitherA_n orAⁿ is referred to as thedegree (ordimension). 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 theprefixco-. In this article, definitions will be given for chain complexes when the distinction is not required.

Abounded chain complex is one in whichalmost all theA_n are 0; that is, a finite complex extended to the left and right by 0. An example is the chain complex defining thesimplicial homology of a finitesimplicial complex. A chain complex isbounded above if all modules above some fixed degreeN are 0, and isbounded 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 ofd are called(co)cycles (orclosed elements), and the elements in the image ofd are called(co)boundaries (orexact elements). Right from the definition of the differential, all boundaries are cycles. Then-th (co)homology groupH_n (Hⁿ) is the group of (co)cyclesmodulo (co)boundaries in degreen, that is,

${\displaystyle H_n=\ker d_n/\text{im }{d}_{n+1}\left(H^n=\ker d^n/\text{im }{d}^{n-1}\right)}$

Anexact sequence (orexact complex) is a chain complex whose homology groups are all zero. This means all closed elements in the complex are exact. Ashort exact sequence is a bounded exact sequence in which only the groupsA_k,A_k+1,A_k+2 may be nonzero. For example, the following chain complex is a short exact sequence.

${\displaystyle \cdots \stackrel{}{\to }0\stackrel{}{\to }\mathbf{Z}\stackrel{\times p}{\to }\mathbf{Z}\twoheadrightarrow \mathbf{Z}/p\mathbf{Z}\stackrel{}{\to }0\stackrel{}{\to }\cdots }$

In the middle group, the closed elements are the elements pZ; these are clearly the exact elements in this group.

Achain mapf between two chain complexes${\displaystyle (A_{\bullet },d_{A,\bullet })}$ and${\displaystyle (B_{\bullet },d_{B,\bullet })}$ is a sequence${\displaystyle f_{\bullet }}$ of homomorphisms${\displaystyle f_n:A_n\to B_n}$ for eachn that commutes with the boundary operators on the two chain complexes, so${\displaystyle d_{B,n}\circ f_n=f_{n-1}\circ d_{A,n}}$. This is written out in the followingcommutative diagram.

A chain map sends cycles to cycles and boundaries to boundaries, and thus induces a map on homology${\displaystyle (f_{\bullet }{)}_{\ast }:H_{\bullet }(A_{\bullet },d_{A,\bullet })\to H_{\bullet }(B_{\bullet },d_{B,\bullet })}$.

A continuous mapf between topological spacesX andY induces a chain map between the singular chain complexes ofX andY, and hence induces a mapf_* between the singular homology ofX andY as well. WhenX andY are both equal to then-sphere, the map induced on homology defines thedegree of the mapf.

The concept of chain map reduces to the one of boundary through the construction of thecone of a chain map.

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 complexesA andB, and two chain mapsf,g :A →B, achain homotopy is a sequence of homomorphismsh_n :A_n →B_n+1 such thathd_A +d_Bh =f −g. The maps may be written out in a diagram as follows, but this diagram is not commutative.

The maphd_A +d_Bh is easily verified to induce the zero map on homology, for anyh. It immediately follows thatf andg induce the same map on homology. One saysf andg arechain homotopic (or simplyhomotopic), and this property defines anequivalence relation between chain maps.

LetX andY be topological spaces. In the case of singular homology, ahomotopy between continuous mapsf,g :X →Y induces a chain homotopy between the chain maps corresponding tof andg. This shows that two homotopic maps induce the same map on singular homology. The name "chain homotopy" is motivated by this example.

LetX be a topological space. DefineC_n(X) fornaturaln to be thefree abelian group formally generated bysingular n-simplices inX, and define the boundary map${\displaystyle {\mathrm{\partial }}_n:C_n(X)\to C_{n-1}(X)}$ to be

${\displaystyle {\mathrm{\partial }}_n:(\sigma :[v_0,\dots ,v_n]\to X)\mapsto (\sum _{i=0}^n(-1{)}^i\sigma :[v_0,\dots ,{\hat{v}}_i,\dots ,v_n]\to X)}$

where the hat denotes the omission of avertex. 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${\displaystyle (C_{\bullet },{\mathrm{\partial }}_{\bullet })}$ is a chain complex; thesingular homology${\displaystyle H_{\bullet }(X)}$ is the homology of this complex.

Singular homology is a useful invariant of topological spaces up tohomotopy equivalence. The degree zero homology group is a free abelian group on thepath-components ofX.

Thedifferentialk-forms on anysmooth manifoldM form arealvector space called Ω^k(M) under addition. Theexterior derivatived maps Ω^k(M) to Ω^k+1(M), andd² = 0 follows essentially fromsymmetry of second derivatives, so the vector spaces ofk-forms along with the exterior derivative are a cochain complex.

${\displaystyle 0\stackrel{\subset }{\to }{\mathrm{\Re }}^c\stackrel{\subset }{\to }{\mathrm{\Omega }}^0(M)\stackrel{d}{\to }{\mathrm{\Omega }}^1(M)\stackrel{d}{\to }{\mathrm{\Omega }}^2(M)\stackrel{d}{\to }{\mathrm{\Omega }}^3(M)\to \cdots }$

The cohomology of this complex is called thede Rham cohomology ofM.Locally constant functions are designated with its isomorphism${\displaystyle {\mathrm{\Re }}^c}$ with c the count of mutually disconnected components ofM. This way the complex was extended to leave the complex exact at zero-form level using the subset operator.

Smooth maps^{[broken anchor]} between manifolds induce chain maps, and smooth homotopies between maps induce chain homotopies.

Chain complexes ofK-modules with chain maps form acategory Ch_K, whereK is a commutative ring.

IfV =V${\displaystyle {}_{\ast }}$ andW =W${\displaystyle {}_{\ast }}$ are chain complexes, theirtensor product${\displaystyle V\otimes W}$ is a chain complex with degreen elements given by

${\displaystyle (V\otimes W{)}_n=\underset{\{i,j|i+j=n\}}{⨁}{V}_i\otimes W_j}$

and differential given by

${\displaystyle \mathrm{\partial }(a\otimes b)=\mathrm{\partial }a\otimes b+(-1{)}^{\left|a\right|}a\otimes \mathrm{\partial }b}$

wherea andb are any two homogeneous vectors inV andW respectively, and${\displaystyle \left|a\right|}$ denotes the degree ofa.

This tensor product makes the category Ch_K into asymmetric monoidal category. The identity object with respect to this monoidal product is the base ringK viewed as a chain complex in degree 0. Thebraiding is given on simple tensors of homogeneous elements by

${\displaystyle a\otimes b\mapsto (-1{)}^{\left|a\right|\left|b\right|}b\otimes a}$

The sign is necessary for the braiding to be a chain map.

Moreover, the category of chain complexes ofK-modules also hasinternal Hom: given chain complexesV andW, the internal Hom ofV andW, denoted Hom(V,W), is the chain complex with degreen elements given by${\displaystyle {\mathrm{\Pi }}_i{\text{Hom}}_K(V_i,W_{i+n})}$ and differential given by

${\displaystyle (\mathrm{\partial }f)(v)=\mathrm{\partial }(f(v))-(-1{)}^{\left|f\right|}f(\mathrm{\partial }(v))}$.

We have anatural isomorphism

${\displaystyle \text{Hom}(A\otimes B,C)\cong \text{Hom}(A,\text{Hom}(B,C))}$

• Amitsur complex
• A complex used to defineBloch's higher Chow groups
• Buchsbaum–Rim complex
• Čech complex
• Cousin complex
• Eagon–Northcott complex
• Gersten complex
• Graph complex^[1]
• Koszul complex
• Moore complex
• Schur complex

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

• Bott, Raoul; Tu, Loring W. (1982),Differential Forms in Algebraic Topology, Berlin, New York:Springer-Verlag,ISBN 978-0-387-90613-3
• Hatcher, Allen (2002).Algebraic Topology. Cambridge:Cambridge University Press.ISBN 0-521-79540-0.

• Homological algebra
• Differential topology

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