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

A chain complex is a sequence of abelian groups or modules C < sub > 0 </ sub >, C < sub > 1 </ sub >, C < sub > 2 </ sub >, ... connected by homomorphisms which are called boundary operators.

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