# 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*. Define a simplicial complex , called the nerve of the covering, as follows:

- There is one vertex for each element of .
- There is one edge for each pair such that .
- In general, there is one
*k*-simplex for each*k+1*-element subset of for which .

Geometrically, the nerve is essentially a "dual complex" (in the sense of a dual graph, or Poincaré duality) for the covering .

The idea of Čech cohomology is that, if we choose a "nice" 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, for which every open set and every finite intersection of open sets is contractible. 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 be a topological space, and let be a presheaf of abelian groups on . Let be an open cover of .

### Simplex

A *q*-**simplex** of is an ordered collection of sets chosen from , such that the intersection of all these sets is non-empty. This intersection is called the *support* of and is denoted .

Now let be such a *q*-simplex. The *j-th partial boundary* of is defined to be the *(q-1)*-simplex obtained by removing the *j*-th set from , that is:

The *boundary* of is defined as the alternating sum of the partial boundaries:

### Cochain

A *q*-**cochain** of with coefficients in is a map which associates to 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 Template:H:title to

The coboundary operator is also sometimes called the codifferential.Template:Fact

#### Cocycle

A *q*-cochain is called a *q*-cocycle if it is in the kernel of δ, hence is the set of all *q*-cocycles.

Thus a (q-1)-cochain *f* is a cocycle if for all *q*-simplices σ the cocycle condition holds. In particular, a 1-cochain *f* is a 1-cocycle if

#### Coboundary

A *q*-cochain is called a *q*-coboundary if it is in the image of *δ* and is the set of all *q*-coboundaries.

For example, a 1-cochain *f* is a 1-coboundary if there exists a 0-cochain *h* such that

### Cohomology

The **Čech cohomology** of with values in is defined to be the cohomology of the cochain complex . Thus the *q*th 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 {ρ_{i}} 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" (*i.e.* all the sets *U*_{α} are contractible to a point, and all finite intersections of sets in are either empty or contractible to a point), then
is isomorphic to the de Rham cohomology.

If *X* is compact Hausdorff, then Čech cohomology (with coefficients in a discrete group) 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 (with respect to the chosen topology) 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 sheaf cohomology to Čech 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 etale topology, the two cohomologies agree for any sheaf, provided that any finite set of points in the base scheme *X* are contained in some open affine subscheme. This is satisfied, for example, if *X* is quasi-projective over an affine scheme.^{[2]}

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 *K*_{n} together with boundary and degeneracy maps. Applying a sheaf *F* to *K*_{∗} yields a simplicial abelian group *F*(*K*_{∗}) whose *n*-th cohomology group is denoted *H*^{n}(*F*(*K*_{∗})). (This group is the same as in case *K* equals .) Then, it can be shown that there is a canonical isomorphism

where the colimit now runs over all hypercoverings.^{[3]}

## References

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|CitationClass=book }} ISBN 0-387-90419-0. ISBN 3-540-90419-0. Chapter 2 Appendix A