# Cohomology group

of a cochain complex $K ^ { . } = ( K ^ {n} , d _ {n} )$ of Abelian groups

The graded group $H ^ { . } ( K) = \oplus H ^ {n} ( K)$, where $H ^ {n} ( K) = \mathop{\rm Ker} d _ {n+} 1 / \mathop{\rm Im} d _ {n}$( see Complex). The group $H ^ {n} ( K)$ is called the $n$- dimensional, or the $n$- th, cohomology group of the complex $K ^ { . }$. This concept is dual to that of homology group of a chain complex (see Homology of a complex).

In the category of modules, the cohomology module of a cochain complex is also called a cohomology group.

The cohomology group of a chain complex $K _ {. } = ( K _ {n} , d _ {n} )$ of $\Lambda$- modules with coefficients, or values, in $A$, where $\Lambda$ is an associative ring with identity and $A$ is a $\Lambda$- module, is the cohomology group

$$H ^ { . } ( K _ {. } , A ) = \oplus H ^ {n} ( K _ {. } , A )$$

of the cochain complex

$$\mathop{\rm Hom} _ \Lambda ( K _ {. } , A ) = \ ( \mathop{\rm Hom} _ \Lambda ( K _ {n} , A ) , d _ {n} ^ {*} ) ,$$

where $d _ {n} ^ {*} ( \gamma ) = \gamma \circ d _ {n}$, $\gamma \in \mathop{\rm Hom} ( K _ {n} , A )$. A special case of this construction is the cohomology group of a polyhedron, the singular cohomology group of a topological space, and the cohomology groups of groups, algebras, etc.

If $0 \rightarrow K _ {. } \rightarrow ^ \alpha L _ { . } \rightarrow ^ \beta M _ {. } \rightarrow 0$ is an exact sequence of complexes of $\Lambda$- modules, where the images of the $K _ {n}$ are direct factors in $L _ {n}$, the following exact sequence arises in a natural way:

$${} \dots \rightarrow H ^ {n} ( M _ {. } , A ) \rightarrow ^ { {\alpha ^ {*}} } \ H ^ {n} ( L _ { . } , A ) \rightarrow ^ { {\beta ^ {*}} } \ H ^ {n} ( K _ {. } , A ) \rightarrow ^ { {d ^ {*}} }$$

$$\rightarrow ^ { {d ^ {*}} } H ^ {n+} 1 ( M _ {. } , A ) \rightarrow \dots .$$

On the other hand, if $K _ {. }$ is a complex of $\Lambda$- modules, and all $K _ {n}$ are projective, then with every exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ of $\Lambda$- modules is associated an exact sequence of cohomology groups:

$${} \dots \rightarrow H ^ {n} ( K _ {. } , A ) \rightarrow H ^ {n} ( K _ {. } ,\ B ) \rightarrow H ^ {n} ( K _ {. } , C ) \rightarrow$$

$$\rightarrow \ H ^ {n+} 1 ( K _ {. } , A ) \rightarrow \dots .$$

See Homology group; Cohomology (for the cohomology group of a topological space).

#### References

 [1] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) [2] R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958) [3] S. MacLane, "Homology" , Springer (1963)