# Chain complex

(Redirected from Cochain 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 the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels.

A cochain complex is similar to a chain complex, except that its homomorphisms follow a different convention. 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 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 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.

## Formal definition

A chain complex ${\displaystyle (A_{\bullet },d_{\bullet })}$ is a sequence of abelian groups or modules ..., A0, A1, A2, A3, A4, ... connected by homomorphisms (called boundary operators or differentials) dn : AnAn−1, such that the composition of any two consecutive maps is the zero map. Explicitly, the differentials satisfy dndn+1 = 0, or with indices suppressed, d2 = 0. The complex may be written out as follows.

${\displaystyle \cdots {\xleftarrow {d_{0}}}A_{0}{\xleftarrow {d_{1}}}A_{1}{\xleftarrow {d_{2}}}A_{2}{\xleftarrow {d_{3}}}A_{3}{\xleftarrow {d_{4}}}A_{4}{\xleftarrow {d_{5}}}A_{5}{\xleftarrow {d_{6}}}\cdots }$

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

${\displaystyle \cdots {\xrightarrow {d^{-1}}}A^{0}{\xrightarrow {d^{0}}}A^{1}{\xrightarrow {d^{1}}}A^{2}{\xrightarrow {d^{2}}}A^{3}{\xrightarrow {d^{3}}}A^{4}{\xrightarrow {d^{4}}}A^{5}{\xrightarrow {d^{5}}}\cdots }$

The index n in either An or An 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.

A bounded chain complex is one in which almost all the An 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 is the group of (co)cycles modulo (co)boundaries in degree n.

### Chain maps and tensor product

There is a natural notion of a morphism between chain complexes called a chain map. Given two complexes M${\displaystyle {}_{*}}$ and N${\displaystyle {}_{*}}$, a chain map between the two is a series of homomorphisms from Mi to Ni such that the entire diagram involving the boundary maps of M and N commutes. Chain complexes with chain maps form a category.

If V = V${\displaystyle {}_{*}}$ and W = W${\displaystyle {}_{*}}$ are chain complexes, their tensor product ${\displaystyle V\otimes W}$ is a chain complex with degree i elements given by

${\displaystyle (V\otimes W)_{i}=\bigoplus _{\{j,k|j+k=i\}}V_{j}\otimes W_{k}}$

and differential given by

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

where a and b are any two homogeneous vectors in V and W respectively, and ${\displaystyle \left|a\right|}$ denotes the degree of a.

This tensor product makes the category ${\displaystyle {\text{Ch}}_{K}}$ (for any commutative ring K) of chain complexes of K-modules 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 braiding 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 of K-modules also has 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 ${\displaystyle \Pi _{i}{\text{Hom}}_{K}(V_{i},W_{i+n})}$ and differential given by

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

We have a natural isomorphism

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

## Examples

### Singular homology

Suppose we are given a topological space X.

Define Cn(X) for natural n to be the free abelian group formally generated by singular n-simplices in X, and define the boundary map

${\displaystyle \partial _{n}:C_{n}(X)\to C_{n-1}(X):\,(\sigma :[v_{0},\ldots ,v_{n}]\to X)\mapsto (\partial _{n}\sigma =\sum _{i=0}^{n}(-1)^{i}(\sigma :[v_{0},\ldots ,{\hat {v}}_{i},\ldots ,v_{n}]\to X),}$

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

${\displaystyle H_{n}(X)=\ker \partial _{n}/{\mbox{im }}\partial _{n+1}.}$

### de Rham cohomology

The differential k-forms on any smooth manifold M form an abelian group (in fact an R-vector space) called Ωk(M) under addition. The exterior derivative dk maps Ωk(M) to Ωk+1(M), and d2 = 0 follows essentially from symmetry of second derivatives, so the vector spaces of k-forms along with the exterior derivative are a cochain complex:

${\displaystyle \Omega ^{0}(M)\ {\stackrel {d_{0}}{\to }}\ \Omega ^{1}(M)\to \Omega ^{2}(M)\to \Omega ^{3}(M)\to \cdots .}$

The cohomology of this complex is called the de Rham cohomology:

${\displaystyle H_{\mathrm {DR} }^{0}(M)=\ker d_{0}=}$ {locally constant functions on M with values in R} ${\displaystyle \cong \mathbb {R} }$#{connected pieces of M}
${\displaystyle H_{\mathrm {DR} }^{k}(M)=\ker d_{k}/\mathrm {im} \,d_{k-1}.}$

## Chain maps

A chain map f between two chain complexes ${\displaystyle (A_{\bullet },d_{A,\bullet })}$ and ${\displaystyle (B_{\bullet },d_{B,\bullet })}$ is a sequence ${\displaystyle f_{\bullet }}$ of module homomorphisms ${\displaystyle f_{n}:A_{n}\rightarrow B_{n}}$ for each n that commutes with the boundary operators on the two chain complexes: ${\displaystyle d_{B,n}\circ f_{n}=f_{n-1}\circ d_{A,n}}$. Such a map sends cycles to cycles and boundaries to boundaries, and thus descends to a map on homology:${\displaystyle (f_{\bullet })_{*}:H_{\bullet }(A_{\bullet },d_{A,\bullet })\rightarrow H_{\bullet }(B_{\bullet },d_{B,\bullet })}$.

A continuous map of topological spaces induces chain maps in both the singular and de Rham chain complexes described above (and in general for the chain complex defining any homology theory of topological spaces) and thus a continuous map induces a map on homology. Because the map induced on a composition of maps is the composition of the induced maps, these homology theories are functors from the category of topological spaces with continuous maps to the category of abelian groups with group homomorphisms.

It is worth noticing that the concept of chain map reduces to the one of boundary through the construction of the cone of a chain map.

## Chain homotopy

Chain homotopies give an important equivalence relation between chain maps. Chain homotopic chain maps induce the same maps on homology groups. A particular case is that homotopic maps between two spaces X and Y induce the same maps from homology of X to homology of Y. Chain homotopies have a geometric interpretation; it is described, for example, in the book of Bott and Tu. See Homotopy category of chain complexes for further information.