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Cochain complex

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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: $${\rm im \hspace{3pt}} d / \ker d.$$