 Chain coefficient

In a based free chain group, write a chain uniquely as a finite sumThe scalars $a_{\sigma }$ are its chain coefficients relative to the chosen basis. For simplicial chains, the basis consists of chosen oriented simplexes; reversing an orientation of a simplex changes its coefficient's sign. For singular chains, the basis elements are singular simplices.

For example, $3[v_0{v}_1]-2[v_1{v}_2]$ has edge coefficients $3$ and $-2$. Applying the boundary operator gives $-3[v_0]+5[v_1]-2[v_2]$. A general chain complex may contain nonfree modules, so a simplex-basis expansion is an additional feature of these models, rather than part of the abstract definition of a chain complex.