= Cochain differential
{title2=$d^{n+1}d^n=0$}
A cochain differential is a degree-one map in a <cochain complex>:
$$
d^n:C^n\longrightarrow C^{n+1},\qquad d^{n+1}\circ d^n=0.
$$
The square-zero identity makes every coboundary a cocycle, so the quotient $H^n(C)=\ker d^n/\operatorname{im}d^{n-1}$ 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.
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