 Cochain differential

A cochain differential is a degree-one map in a cochain complex:The square-zero identity makes every coboundary a cocycle, so the quotient $H^n(C)=\ker d^n/\mathrm{im}{d}^{n-1}$ defining cohomology is meaningful. The maps are homomorphisms of the abelian groups, or module-linear maps, used in the complex.

For $C^0=\mathbb{Z}$, $C^1=\mathbb{Z}$, $d^0(n)=2n$, and all other groups zero, the identity is automatic. The degree-zero cohomology is zero, while the degree-one cohomology is $\mathbb{Z}/2\mathbb{Z}$: every degree-one element is closed, but only the even ones are exact.

The exterior derivative on differential forms is another cochain differential, with $d^2=0$. In contrast, a chain differential lowers degree. The word differential records this graded square-zero structure, not necessarily differentiation of scalar functions.