- Hyperhomology
In
homological algebra , the hyperhomology or hypercohomology of a complex of objects of anabelian category is an extension of the usual homology of an object to complexes.It is a sort of cross between the derived functor cohomology of an object and the homology of a chain complex.Hyperhomology is no longer used much: since about 1970 it has been largely replaced by the roughly equivalent concept of a
derived functor betweenderived categories .Definition
We give the definition for hypercohomology as this is more common. As usual, hypercohomology and hyperhomology are essentially the same: one converts from one to the other by changing the direction of all arrows, replacing injective objects to projective ones, and so.
Suppose that "A" is an abelian category with
enough injective objects and "F" aleft exact functor to another abelian category. If "C" is a complex of objects of "A" bounded on the left, the hypercohomology:H"i"("C")
of "C" (for an integer "i") iscalculated as follows:
# Take aquasi-isomorphism "Φ" : "C" -> "I", here "I" is a complex of injective elements of "A"
# The hypercohomology H"i"("C") of "C" is then the cohomology "H""i"("F"("I")) of the complex "F"("I").The hypercohomology of "C" is independent of the choice of the
quasi-isomorphism , up to unique isomorphisms.The hypercohomology can also be defined using derived categories: the hypercohomology of "C" is just the homology of "C" considered as an element of the derived category of "A".
The hypercohomology spectral sequences
There are two hypercohomology
spectral sequence s; one with "E"2 term:"H""i"("R""j""F"("C"))
and the other with "E"2 term
:"R""j""F"("H""i"("C"))
both converging to the hypercohomology
:H"i"+"j"("C"),
where "R""j""F" is a
right derived functor of "F".References
*H. Cartan, S. Eilenberg, "Homological algebra" ISBN 0691049912
*springer|id=H/h048480|title=Hyperhomology functor|author=V.I. Danilov
* A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohoku Math. J. 9 (1957) pp. 119-221
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