Chain complexes

A fundamental intuitive fact reproduced in this formalism is that the boundary of a boundary is zero. A useful algebraic generalization of this idea is the chain complex, defined to be a sequence of homomorphisms of abelian groups \({\partial_{n}\colon C_{n}\to C_{n-1}}\) with \({\partial_{n}\partial_{n+1}=0}\). In our case the abelian groups \({C_{n}}\) are the \({n}\)-chains, and the chain complex can be illustrated as follows:

26.chain-complex

An Illustrated Handbook