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A finite simplicial complex is a simplicial complex containing only finitely many simplexes. With the convention that every vertex belongs to a simplex, this is equivalent to having finitely many vertices of a simplicial complex: a finite vertex set has only finitely many subsets.

Its geometric realization is a finite union of compact geometric simplexes, hence compact. The simplicial chain groups have finite bases, so their boundary maps can be represented by finite matrices. This makes simplicial homology a finite linear-algebra calculation after coefficients are chosen.

Three vertices and their three joining edges form a finite complex whose realization is a circle. Adding the filled triangular face produces a finite complex realizing a disk, which is contractible. Thus finiteness makes the data manageable but does not determine the homotopy type.

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.

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]