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 contained in the kernel of the next. Associated to a chain complex is its homology, which is (loosely speaking) a measure of the failure of a chain complex to be exact.

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 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.

A chain complex ${\displaystyle (A_{\bullet },d_{\bullet })}$ is a sequence of abelian groups or modules ${\displaystyle \cdots ,A_{0},A_{1},A_{2},\dots }$ connected by homomorphisms (called boundary operators or differentials) ${\displaystyle d_{n}:A_{n}\to A_{n-1}}$, such that the composition of any two consecutive maps is the zero map. Explicitly, the differentials satisfy ${\displaystyle d_{n}\circ d_{n+1}=0}$ for all ${\displaystyle n}$, or, concisely, ${\displaystyle d^{2}=0}$. The complex may be written out as follows:

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 ${\displaystyle \cdots ,A^{0},A^{1},A^{2},\dots }$ connected by homomorphisms (coboundary operators) ${\displaystyle d^{n}:A^{n}\to A^{n+1}}$ satisfying ${\displaystyle d^{n+1}\circ d^{n}=0}$. The cochain complex may be written out in a similar fashion to the chain complex:

In both cases, the index ${\displaystyle n}$ 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 the ${\displaystyle A_{n}}$ 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 ${\displaystyle N}$ 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.