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.
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 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
- H^{i}(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 H^{i}(C) of C is then the cohomology H^{i}(F(I)) of the complex F(I).
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 functors of H^{0} = FH^{0} = H^{0}F.
The hypercohomology spectral sequences
There are two hypercohomology spectral sequences; one with E_{2} term
and the other with E_{1} term
and E_{2} term
both converging to the hypercohomology
- ,
where R^{j}F is a right derived functor of F.
Examples
- For a variety X over a field k, the second spectral sequence from above gives the Hodge to de Rham spectral sequence for algebraic de Rham cohomology:
- .
- Another example comes from the holomorphic log complex on a complex manifold. Let X be a complex algebraic manifold and a good compactification. This means that Y is a compact algebraic manifold and is a divisor on with simple normal crossings. The natural inclusion of complexes of sheaves
turns out to be a quasi-isomorphism and induces an isomorphism
- .
See also
References
- H. Cartan, S. Eilenberg, Homological algebra ISBN 0-691-04991-2
- V.I. Danilov (2001) [1994], "Hyperhomology functor", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- A. Grothendieck, Sur quelques points d'algèbre homologique Tohoku Math. J. 9 (1957) pp. 119-221