Chain complex

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In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or an algebraic construction such as a simplicial complex. More generally, homological algebra includes the study of chain complexes in the abstract, without any reference to an underlying space. In this case, chain complexes are studied axiomatically as algebraic structures.

Applications of chain complexes usually define and apply their homology groups (cohomology groups for cochain complexes); in more abstract settings various equivalence relations are applied to complexes (for example starting with the chain homotopy idea). Chain complexes are easily defined in abelian categories, also.


Formal definition

A chain complex (A_\bullet, d_\bullet) is a sequence of abelian groups or modules ... A2, A1, A0, A-1, A-2, ... connected by homomorphisms (called boundary operators) dn : AnAn−1, such that the composition of any two consecutive maps is zero: dndn+1 = 0 for all n. They are usually written out as:

A variant on the concept of chain complex is that of cochain complex. A cochain complex (A^\bullet, d^\bullet) is a sequence of abelian groups or modules ..., A − 2, A − 1, A0, A1, A2, ... connected by homomorphisms d^n\colon A^n \to A^{n + 1} such that the composition of any two consecutive maps is zero: dn + 1dn = 0 for all n:

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