Editing Chain complex (section)

{{Short description|Tool in homological algebra}}
In [[mathematics]], a '''chain complex''' is an [[algebraic structure]] that consists of a sequence of [[abelian group]]s (or [[module (mathematics)|modules]]) and a sequence of [[group homomorphism|homomorphisms]] between consecutive groups such that the [[image (mathematics)|image]] of each homomorphism is contained in the [[kernel (algebra)#Group homomorphisms|kernel]] of the next. Associated to a chain complex is its [[Homology (mathematics)|homology]], which is (loosely speaking) a measure of the failure of a chain complex to be [[Exact sequence|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 function#Continuous functions between topological spaces|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 [[topological invariant|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]].