Definition:Cohomology Groups

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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 $\struct {M, d}$.