Chain complexes

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.

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) d_n : A_n→A_n−1, such that the composition of any two consecutive maps is the zero map. Explicitly, the differentials satisfy d_n ∘ d_n+1 = 0, or with indices suppressed, d² = 0. The complex may be written out as follows.

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