# Definition:Cohomology Groups

## Definition

Let $\left({M, d}\right)$ be a differential complex with grading:

$\displaystyle M = \bigoplus_{n \in \Z} M^n$

Let $d_n := d \restriction_{M_n}$.

Elements of the module $M$ are called cochains.

Elements of the submodule $Z^n \left({M}\right) = \ker \left({d_n}\right)$ are called cocycles.

Elements of the submodule $B^n \left({M}\right) = \operatorname{Im} \left({d_{n-1}}\right)$ are called coboundaries.

The modules (and hence groups) $H^n \left({M}\right) = Z^n \left({M}\right)/B^n \left({M}\right)$ are called the cohomology groups of the differential complex $\left({M, d}\right)$.