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Homology functor

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A functor on an Abelian category that defines a certain homological structure on it. A system of covariant additive functors from an Abelian category {\mathcal A} into an Abelian category {\mathcal A} _ {1} is called a homology functor if the following axioms are satisfied.

1) For each exact sequence

0 \rightarrow A ^ \prime \rightarrow A \rightarrow A ^ {\prime\prime} \rightarrow 0

and each i , in {\mathcal A} a morphism \partial _ {i} : H _ {i+ 1} ( A ^ {\prime\prime} ) \rightarrow H _ {i} ( A ^ \prime ) is given, which is known as the connecting or boundary morphism.

2) The sequence

\dots \rightarrow H _ { i + 1 } ( A ^ \prime ) \rightarrow H _ {i + 1 } ( A) \rightarrow \ H _ {i + 1 } ( A ^ {\prime\prime} ) \rightarrow ^ { {\partial _ i } }

\rightarrow ^ { {\partial _ i} } H _ {i} ( A ^ \prime ) \rightarrow \dots ,

called the homology sequence, is exact.

Thus, let {\mathcal A} = K( \mathop{\rm Ab} ) be the category of chain complexes of Abelian groups, and let \mathop{\rm Ab} be the category of Abelian groups. The functors H _ {i} : K( \mathop{\rm Ab} ) \rightarrow \mathop{\rm Ab} which assign to a complex K _ {\mathbf . } the corresponding homology groups H _ {i} ( K _ {\mathbf . } ) define a homology functor.

Let F: {\mathcal A} \mapsto {\mathcal A} _ {1} be an additive covariant functor for which the left derived functors R _ {i} F ( R _ {i} F = 0 , i < 0 ) are defined (cf. Derived functor). The system ( R _ {i} F ) _ {i \in \mathbf Z } will then define a homology functor from {\mathcal A} into {\mathcal A} _ {1} .

Another example of a homology functor is the hyperhomology functor.

A cohomology functor is defined in a dual manner.

References

[1] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohoku Math. J. , 9 (1957) pp. 119–221
How to Cite This Entry:
Homology functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homology_functor&oldid=52353
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article