In homological algebra, the hyperhomology or hypercohomology (
) is a generalization of (co)homology functors which takes as input not objects in an abelian category
It is a sort of cross between the derived functor cohomology of an object and the homology of a chain complex since hypercohomology corresponds to the derived global sections functor
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.
One of the motivations for hypercohomology comes from the fact that there isn't an obvious generalization of cohomological long exact sequences associated to short exact sequences
i.e. there is an associated long exact sequence
It turns out hypercohomology gives techniques for constructing a similar cohomological associated long exact sequence from an arbitrary long exact sequence
since its inputs are given by chain complexes instead of just objects from an abelian category.
We can turn this chain complex into a distinguished triangle (using the language of triangulated categories on a derived category)
Then, taking derived global sections
gives a long exact sequence, which is a long exact sequence of hypercohomology groups.
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 RF(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 functors 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 RjF is a right derived functor of F. One application of hypercohomology spectral sequences are in the study of gerbes.
Recall that rank n vector bundles on a space
can be classified as the Cech-cohomology group
, we instead consider the cohomology group
, so it classifies objects which are glued by objects in the original classifying group.
A closely related subject which studies gerbes and hypercohomology is Deligne-cohomology.
turns out to be a quasi-isomorphism and induces an isomorphism