Čech cohomology


In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.

Motivation

Let X be a topological space, and let be an open cover of X. Let denote the nerve of the covering. The idea of Čech cohomology is that, for an open cover consisting of sufficiently small open sets, the resulting simplicial complex should be a good combinatorial model for the space X. For such a cover, the Čech cohomology of X is defined to be the simplicial cohomology of the nerve. This idea can be formalized by the notion of a good cover. However, a more general approach is to take the direct limit of the cohomology groups of the nerve over the system of all possible open covers of X, ordered by refinement. This is the approach adopted below.

Construction

Let X be a topological space, and let be a presheaf of abelian groups on X. Let be an open cover of X.

Simplex

A q-simplex σ of is an ordered collection of q+1 sets chosen from, such that the intersection of all these sets is non-empty. This intersection is called the support of σ and is denoted |σ|.
Now let be such a q-simplex. The j-th partial boundary of σ is defined to be the -simplex obtained by removing the j-th set from σ, that is:
The boundary of σ is defined as the alternating sum of the partial boundaries:
viewed as an element of the free abelian group spanned by the simplices of.

Cochain

A q-cochain of with coefficients in is a map which associates with each q-simplex σ an element of and we denote the set of all q-cochains of with coefficients in by. is an abelian group by pointwise addition.

Differential

The cochain groups can be made into a cochain complex by defining the coboundary operator by:
where is the restriction morphism to
A calculation shows that
The coboundary operator is analogous to the exterior derivative of De Rham cohomology, so it sometimes called
the differential of the cochain complex.

Cocycle

A q-cochain is called a q-cocycle if it is in the kernel of, hence is the set of all q-cocycles.
Thus a -cochain is a cocycle if for all q-simplices the cocycle condition
holds.
A 0-cocycle is a collection of local sections of satisfying a compatibility relation on every intersecting
A 1-cocycle satisfies for every non-empty with

Coboundary

A q-cochain is called a q-coboundary if it is in the image of and is the set of all q-coboundaries.
For example, a 1-cochain is a 1-coboundary if there exists a 0-cochain such that for every intersecting

Cohomology

The Čech cohomology of with values in is defined to be the cohomology of the cochain complex. Thus the qth Čech cohomology is given by
The Čech cohomology of X is defined by considering refinements of open covers. If is a refinement of then there is a map in cohomology The open covers of X form a directed set under refinement, so the above map leads to a direct system of abelian groups. The Čech cohomology of X with values in ' is defined as the direct limit of this system.
The Čech cohomology of X with coefficients in a fixed abelian group A, denoted, is defined as where is the constant sheaf on X determined by A.
A variant of Čech cohomology, called numerable Čech cohomology', is defined as above, except that all open covers considered are required to be numerable: that is, there is a partition of unity such that each support is contained in some element of the cover. If X'' is paracompact and Hausdorff, then numerable Čech cohomology agrees with the usual Čech cohomology.

Relation to other cohomology theories

If X is homotopy equivalent to a CW complex, then the Čech cohomology is naturally isomorphic to the singular cohomology. If X is a differentiable manifold, then is also naturally isomorphic to the de Rham cohomology; the article on de Rham cohomology provides a brief review of this isomorphism. For less well-behaved spaces, Čech cohomology differs from singular cohomology. For example if X is the closed topologist's sine curve, then whereas
If X is a differentiable manifold and the cover of X is a "good cover", then is isomorphic to the de Rham cohomology.
If X is compact Hausdorff, then Čech cohomology is isomorphic to Alexander-Spanier cohomology.

In algebraic geometry

Čech cohomology can be defined more generally for objects in a site C endowed with a topology. This applies, for example, to the Zariski site or the etale site of a scheme X. The Čech cohomology with values in some sheaf F is defined as
where the colimit runs over all coverings of X. Here is defined as above, except that the r-fold intersections of open subsets inside the ambient topological space are replaced by the r-fold fiber product
As in the classical situation of topological spaces, there is always a map
from Čech cohomology to sheaf cohomology. It is always an isomorphism in degrees n = 0 and 1, but may fail to be so in general. For the Zariski topology on a Noetherian separated scheme, Čech and sheaf cohomology agree for any quasi-coherent sheaf. For the étale topology, the two cohomologies agree for any étale sheaf on X, provided that any finite set of points of X are contained in some open affine subscheme. This is satisfied, for example, if X is quasi-projective over an affine scheme.
The possible difference between Cech cohomology and sheaf cohomology is a motivation for the use of hypercoverings: these are more general objects than the Cech nerve
A hypercovering K∗ of X is a simplicial object in C, i.e., a collection of objects Kn together with boundary and degeneracy maps. Applying a sheaf F to K∗ yields a simplicial abelian group F whose n-th cohomology group is denoted Hn. Then, it can be shown that there is a canonical isomorphism
where the colimit now runs over all hypercoverings.

Examples

For example, we can compute the coherent sheaf cohomology of on the projective line using the Čech complex. Using the cover
we have the following modules from the cotangent sheaf
If we take the conventions that then we get the Čech complex
Since is injective and the only element not in the image of is we get that

Citation footnotes

General references