 Simplicial chain group

For an abstract simplicial complex $K$, the integer simplicial chain group $C_n(K;\mathbb{Z})$ is the free abelian group on its oriented $n$-simplexes, with reversal of orientation changing sign. Thus a chain is a finite formal integer sum of oriented simplices. After choosing one orientation for each simplex, those chosen simplices form a basis of this abelian group.

For a filled triangle with vertices $0,1,2$, $C_0\cong {\mathbb{Z}}^3$, $C_1\cong {\mathbb{Z}}^3$ and $C_2\cong \mathbb{Z}$. Its oriented two-simplex has boundaryTaking another boundary gives $([2]-[1])-([2]-[0])+([1]-[0])=0$. This cancellation makes the simplicial chain complex possible: boundaries are cycles, and simplicial homology measures cycles modulo boundaries. With coefficients in an abelian group $A$, use $C_n(K;\mathbb{Z}){\otimes }_{\mathbb{Z}}A$.^[ref]