In
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, the hyperhomology or hypercohomology (
) is a generalization of (co)homology functors which takes as input not objects in an
abelian category 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.
Motivation
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)
which we denote by
Then, taking derived global sections
gives a long exact sequence, which is a long exact sequence of hypercohomology groups.
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
In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categori ...
and ''F'' a
left exact functor
In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much ...
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
In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism ''A'' → ''B'' of chain complexes (respectively, cochain complexes) such that the induced morphisms
:H_n(A_\bullet) \to H_n(B_\bullet)\ (\text H^n(A^\bull ...
''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
In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism ''A'' → ''B'' of chain complexes (respectively, cochain complexes) such that the induced morphisms
:H_n(A_\bullet) \to H_n(B_\bullet)\ (\text H^n(A^\bull ...
, 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 H
0 = ''FH''
0 = ''H''
0''F''.
The hypercohomology spectral sequences
There are two hypercohomology
spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they hav ...
s; 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
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
Motivation
It was noted in var ...
of ''F''.
Applications
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
. The main idea behind gerbes is to extend this idea cohomologically, so instead of taking
for some functor
, 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.
Examples
See also
*
Cartan–Eilenberg resolution In homological algebra, the Cartan–Eilenberg resolution is in a sense, a resolution of a chain complex. It can be used to construct hyper-derived functors. It is named in honor of Henri Cartan and Samuel Eilenberg.
Definition
Let \mathcal be a ...
*
Gerbe
References
*H. Cartan, S. Eilenberg, ''Homological algebra''
*{{springer, id=H/h048480, title=Hyperhomology functor, author=V.I. Danilov
* A. Grothendieck, ''Sur quelques points d'algèbre homologique'' Tohoku Math. J. 9 (1957) pp. 119-221
Homological algebra