Definition:Cohomology Groups

Definition
Let $(M,d)$ be a differential complex with grading


 * $\displaystyle M = \oplus_{n \in \Z} M^n$

Let $d_n := d\big|_{M_n}$.

Elements of the module $M$ are called cochains

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

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

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