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

Hⁱ(C)

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

1. Take a quasi-isomorphism Φ : C → I, here I is a complex of injective elements of A.
2. The hypercohomology Hⁱ(C) of C is then the cohomology Hⁱ(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 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.

The hypercohomology spectral sequences

There are two hypercohomology spectral sequences; one with E₂ term

${\displaystyle R^{i}F(H^{j}(C))}$

and the other with E₁ term

${\displaystyle R^{j}F(C^{i})}$

and E₂ term

${\displaystyle H^{i}(R^{j}F(C))}$

both converging to the hypercohomology

${\displaystyle H^{i+j}(RF(C))}$,

where R^jF 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-de Rham spectral sequence for algebraic de Rham cohomology:

  ${\displaystyle E_{1}^{p,q}=H^{q}(X,\Omega _{X}^{p})\Rightarrow \mathbf {H} ^{p+q}(X,\Omega _{X}^{\bullet })=:H_{DR}^{p+q}(X/k)}$.

• Another example comes from the holomorphic log complex on a complex manifold. Let X be a complex algebraic manifold and ${\displaystyle j:X\hookrightarrow Y}$ a good compactification. This means that Y is a compact algebraic manifold and ${\displaystyle D=Y-X}$ is a divisor on ${\displaystyle Y}$ with simple normal crossings. The natural inclusion of complexes of sheaves

  ${\displaystyle \Omega _{Y}^{\bullet }(\log D)\rightarrow j_{*}\Omega _{X}^{\bullet }}$

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

  ${\displaystyle H^{k}(X;\mathbb {C} )\rightarrow \mathbf {H} ^{k}(Y,\Omega _{Y}^{\bullet }(\log D))}$.

See also

• Cartan–Eilenberg resolution

References

• H. Cartan, S. Eilenberg, Homological algebra ISBN 0-691-04991-2
• V.I. Danilov (2001) [1994], "Hyperhomology functor", Encyclopedia of Mathematics, EMS Press
• A. Grothendieck, Sur quelques points d'algèbre homologique Tohoku Math. J. 9 (1957) pp. 119-221