Simplicial chain ($c={\sum }_{\sigma }{a}_{\sigma }\sigma$)

A simplicial $q$-chain is a finite formal linear combination of oriented $q$-simplices of a simplicial complex, with coefficients in a specified abelian group or ring. With integer coefficients,where reversing a simplex's orientation changes its sign. The chains form the simplicial chain group. In characteristic two the sign distinction disappears.

The boundary operator is extended linearly fromFor a filled triangle, $\partial [0,1,2]=[1,2]-[0,2]+[0,1]$. Applying the boundary again gives $(2-1)-(2-0)+(1-0)=0$, where each vertex label denotes its basis zero-chain. The cancellation exemplifies ${\partial }^2=0$.

A chain with zero boundary is a simplicial cycle; a chain that is the boundary of a higher chain is a chain boundary. Their quotient is the relevant simplicial homology group.