If $(\mathcal{A},d)$ is a cochain complex $$ \cdots \xrightarrow{d_{n-1}} A^{n-1} \xrightarrow{d_{n}} A^n \xrightarrow {d_{n+1}} A^{n+1} \xrightarrow{d_{n+2}} \cdots $$ then the $n^{\mathrm{th}}$ cohomology group (or cohomology module) $H^n(\mathcal{A},d)$ of $(\mathcal{A},d)$ is the quotient module $$ H^n(\mathcal{A},d)=\frac{\ker d_{n+1}}{\im d_n}. $$

The cochain complex is an exact sequence if and only if all of the cohomology groups are trivial. The cohomology groups can therefore be thought of as measuring the extent to which the cochain complex fails to be exact.

Cohomology groups of other objects are defined as the cohomology groups of an associated cochain complex. (For example, see the entry on the cohomology of simplicial complexes.)

[Compare this entry with the entry on homology of a chain complex.]