Hyperhomology

In homological algebra, the hyperhomology or hypercohomology () is a generalization of (co)homology functors which takes as input not objects in an but instead chain complexes of objects, so objects in . 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 sequencesi.e. there is an associated long exact sequenceIt turns out hypercohomology gives techniques for constructing a similar cohomological associated long exact sequence from an arbitrary long exact sequencesince 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)which we denote byThen, 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 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).