Hyperhomology

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.

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 dualizing, i.e. by changing the direction of all arrows, replacing injective objects with projective ones, and so on.

Suppose that A is an abelian category with enough injectives and F a left exact functor to another abelian category B. If C is a complex of objects of A bounded on the left, the hypercohomology

of C (for an integer i) is calculated as follows:

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 cohomology of F(C) considered as an element of the derived category of B.

For complexes that vanish for negative indices, the hypercohomology can be defined as the derived functions of H⁰ = FH⁰ = H⁰F.

There are two hypercohomology spectral sequences; one with E₂ term

and the other with E₁ term

and E₂ term

both converging to the hypercohomology

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

turns out to be a quasi-isomorphism and induces an isomorphism

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