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)

Comments

The exact sequence of cohomology groups given above is often referred to as a long exact sequence of cohomology groups associated to a short exact sequence of complexes.