\begin{definition}[Definition:Cohomology Groups]
Let $\struct {M, d}$ be a differential complex with grading:
:$\ds M = \bigoplus_{n \mathop \in \Z} M^n$
Let $d_n := d \restriction_{M_n}$.
Elements of the module $M$ are called '''cochains'''.
Elements of the submodule $\map {Z^n} M = \map \ker {d_n}$ are called '''cocycles'''.
Elements of the submodule $\map {B^n} M = \Img {d_{n - 1} }$ are called '''coboundaries'''.
The modules (and hence groups) $\map {H^n} M = \map {Z^n} M / \map {B^n} M$ are called the '''cohomology groups''' of the differential complex $\struct {M, d}$.
Category:Definitions/Homological Algebra
\end{definition}