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 right 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:

1. Take an injective resolution I of C (this means that I is a complex of injective elements of A together with a map of complexes from C to I that is a homotopy equivalence).
2. The hypercohomoloy 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 injective resolution, 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(R^jF(C))

and the other with E₂ term

R^jF(Hⁱ(C))

both converging to the hypercohomology

H^i+j(C),

where R^jF is a right derived functor of F.

[edit] References

• H. Cartan, S. Eilenberg, Homological algebra ISBN 0691049912
• V.I. Danilov, "Hyperhomology functor" SpringerLink Encyclopaedia of Mathematics (2001)
• A. Grothendieck, Sur quelques points d'algèbre homologique Tohoku Math. J. 9 (1957) pp. 119-221