Hyperhomology
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In homological algebra, the hyperhomology or hypercohomology of a complex of objects of an abelian 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 between derived categories.
[edit] 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 a left exact functor to another abelian category. If C is a complex of objects of A bounded on the left, the hypercohomology
- Hi(C)
of C (for an integer i) is calculated as follows:
- Take a quasi-isomorphism Φ : C -> I, here I is a complex of injective elements of A
- The hypercohomology Hi(C) of C is then the cohomology Hi(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.
[edit] The hypercohomology spectral sequences
There are two hypercohomology spectral sequences; one with E2 term
- Hi(RjF(C))
and the other with E2 term
- RjF(Hi(C))
both converging to the hypercohomology
- Hi+j(C),
where RjF is a right derived functor of F.
[edit] References
- H. Cartan, S. Eilenberg, Homological algebra ISBN 0691049912
- V.I. Danilov (2001), “Hyperhomology functor”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- A. Grothendieck, Sur quelques points d'algèbre homologique Tohoku Math. J. 9 (1957) pp. 119-221
