Hyperhomology functor

A series of functors on the category of complexes connected with some functor . In fact, let be a covariant additive functor from an Abelian category with a sufficient number of projective objects into an Abelian category . Further, let be a chain complex with values in and let be a Cartan–Eilenberg resolution of , consisting of projective objects. Then the bicomplex determines the homology groups and two spectral sequences (cf. Spectral sequence) converging to them with initial terms

These homology groups and spectral sequences depend functorially on and are known, respectively, as the hyperhomology functors for and the spectral hyperhomology functors for . The hyperhomology functor is a homology functor on the category of complexes in the following important cases: when commutes with inductive limits; when the objects in the category have projective resolutions of length ; or when it is considered on the category of complexes with positive degrees.

Hypercohomology functors are defined dually.

References

[1]	H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[2]	A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohoku Math. J. , 9 (1957) pp. 119–221