Chain Complex

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

\cdots \to A_{n+1} \xrightarrow{d_{n+1}} A_n \xrightarrow{d_n} A_{n-1} \xrightarrow{d_{n-1}} A_{n-2} \to \cdots \xrightarrow{d_2} A_1 \xrightarrow{d_1} A_0 \xrightarrow{d_0} A_{-1} \xrightarrow{d_{-1}} A_{-2} \xrightarrow{d_{-2}} \cdots

A variant on the concept of chain complex is that of cochain complex. A cochain complex is a sequence of abelian groups or modules ..., ... connected by homomorphisms such that the composition of any two consecutive maps is zero: for all n:

\cdots \to A^{-2} \xrightarrow{d^{-2}} A^{-1} \xrightarrow{d^{-1}} A^0 \xrightarrow{d^0} A^1 \xrightarrow{d^1} A^2 \to \cdots \to A^{n-1} \xrightarrow{d^{n-1}} A^n \xrightarrow{d^n} A^{n+1} \to \cdots.

The index in either or is referred to as the degree (or dimension). The only difference in the definitions of chain and cochain complexes is that, in chain complexes, the boundary operators decrease dimension, whereas in cochain complexes they increase dimension.

A bounded chain complex is one in which almost all the Ai are 0; i.e., a finite complex extended to the left and right by 0's. An example is the complex defining the homology theory of a (finite) simplicial complex. A chain complex is bounded above if all degrees above some fixed degree N are 0, and is bounded below if all degrees below some fixed degree are 0. Clearly, a complex is bounded above and below if and only if the complex is bounded.

Leaving out the indices, the basic relation on d can be thought of as

The elements of the individual groups of a chain complex are called chains (or cochains in the case of a cochain complex.) The image of d is the group of boundaries, or in a cochain complex, coboundaries. The kernel of d (i.e., the subgroup sent to 0 by d) is the group of cycles, or in the case of a cochain complex, cocycles. From the basic relation, the (co)boundaries lie inside the (co)cycles. This phenomenon is studied in a systematic way using (co)homology groups.

Read more about Chain Complex:  Chain Maps, Chain Homotopy

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