Čech cohomology

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A Penrose triangle depicts a nontrivial element of the first cohomology of an annulus with values in the group of distances from the observer[1]

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.

MotivationEdit

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.

ConstructionEdit

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

SimplexEdit

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 (q−1)-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  .

CochainEdit

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.

DifferentialEdit

The cochain groups can be made into a cochain complex   by defining the coboundary operator   by:

 

where   is the restriction morphism from   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.

CocycleEdit

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 (q−1)-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  

 

CoboundaryEdit

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  

 

CohomologyEdit

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 unityi} 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 theoriesEdit

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 Uα 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 geometryEdit

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 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.[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 Kn 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 Hn(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]

ExamplesEdit

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

 

ReferencesEdit

Citation footnotesEdit

  1. ^ Penrose, Roger (1992), "On the Cohomology of Impossible Figures", Leonardo, 25 (3/4): 245–247, doi:10.2307/1575844. Reprinted from Penrose, Roger (1991), "On the Cohomology of Impossible Figures / La Cohomologie des Figures Impossibles", Structural Topology, 17: 11–16, retrieved January 16, 2014
  2. ^ Milne, James S. (1980), Étale cohomology, Princeton Mathematical Series, 33, Princeton University Press, ISBN 978-0-691-08238-7, MR 0559531, Section III.2, Theorem 2.17
  3. ^ Artin, Michael; Mazur, Barry (1969), Etale homotopy, Lecture Notes in Mathematics, No. 100, Berlin, New York: Springer-Verlag, Theorem 8.16

General referencesEdit