In homological algebra, a chain complex is a sequence of abelian groups or modules A₀, A₁, A₂... connected by homomorphisms d_n : A_n→A_n-1, such that the composition of any two consecutive maps is zero: d_n o d_n+1 = 0 for all n. They tend to be written out like so:

A variant on the concept of chain complex is that of cochain complex. A cochain complex is a sequence of abelian groups or modules A₀, A₁, A₂... connected by homomorphisms d_n : A_n→A_n+1, such that the composition of any two consecutive maps is zero: d_n+1 o d_n = 0 for all n:

The idea is basically the same.

Chain complexes are mainly used to define homology and cohomology.

Examples

Singular homology

Suppose we are given a topological space X.

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

where the hat denotes the omission of a vertex. That is, the boundary of a singular simplex is alternating sum of restrictions to its faces. It can be shown ∂² = 0, so is a chain complex; the singular homology is the homology of this complex; that is,

.

de Rham cohomology

The differential k-forms on any smooth manifold M form an abelian group (in fact an R_vector space) called Ω^k(M) under addition. The exterior derivative d = d _k maps Ω^k(M) → Ω^k+1(M), and d ² = 0 follows essentially from symmetry of second derivatives, so the vector spaces of k-forms along with the exterior derivative are a cochain complex:

The homology of this complex is the de Rham cohomology

.

Group cohomology