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# Cochain complex

A *cochain complex* $C^*$ is a sequence of groups and homomorphisms:
$$\ldots\stackrel{d}{\leftarrow}C^{n+1}\stackrel{d}{\leftarrow}C^{n}\stackrel{d}{\leftarrow}\ldots
\stackrel{d}{\leftarrow}C^{0}\stackrel{d}{\leftarrow}0$$
satisfying:
$${\rm im \hspace{3pt}} d \subset \ker d.$$

Here $d$ is called the *coboundary operator*.

It looks like a "reversed" chain complex.

These groups may be the spaces of differential forms, continuous or discrete, with $d$ the exterior derivative, or from cochains of cubical complexes or other cell complexes.

The elements of $\ker d$ are called *cocycles* and elements of ${\rm im \hspace{3pt}} d$ are *coboundaries*.

Whatever the source of the groups, cohomology now is defined as the quotient: $${\rm im \hspace{3pt}} d / \ker d.$$