= Simplicial chain
{title2=$c=\sum_\sigma a_\sigma\sigma$}
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,
$$
c=\sum_\sigma a_\sigma\sigma,\qquad a_\sigma\in\mathbb Z,
$$
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 from
$$
\partial[v_0,\ldots,v_q]=\sum_{j=0}^q(-1)^j[v_0,\ldots,\widehat v_j,\ldots,v_q].
$$
For 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.
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