Chain complex

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 a topological space. 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 is a sequence of abelian groups or modules ... A₂, A₁, A₀, A_-1, A_-2, ... connected by homomorphisms (called boundary operators) d_n : A_n→A_n−1, such that the composition of any two consecutive maps is zero: d_n ∘ d_n+1 = 0 for all n. They are usually written out as:

Ribosome-nascent chain complex

Ribosome-nascent chain complex (RNC) refers to the collection of molecules that constitute a ribosome attached to the polypeptide (protein) that it is synthesising. The synthesis of the nascent polypeptide may be stalled by one of several methods. RNCs are produced and purified in laboratories to study the dynamics, biochemistry, folding and interactions of both the ribosome and proteins undergoing synthesis.

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