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.
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Cech Cohomology Motivation I

Computing Sheaf Cohomology for Products of Projective Spaces

Defn Cech Cohomology

Basic Notions Seminar Series: An introduction to cohomology, Speaker: Ben Mares

Cech Cohomology Motivation 2
Transcription
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 qsimplex σ of is an ordered collection of q+1 sets chosen from , such that the intersection of all these sets is nonempty. This intersection is called the support of σ and is denoted σ.
Now let be such a qsimplex. The jth partial boundary of σ is defined to be the (q−1)simplex obtained by removing the jth 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 qcochain of with coefficients in is a map which associates with each qsimplex σ an element of , and we denote the set of all qcochains 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 from to (Notice that ∂_{j}σ ⊆ σ, but σ ⊆ ∂_{j}σ.)
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 qcochain is called a qcocycle if it is in the kernel of , hence is the set of all qcocycles.
Thus a (q−1)cochain is a cocycle if for all qsimplices the cocycle condition
holds.
A 0cocycle is a collection of local sections of satisfying a compatibility relation on every intersecting
A 1cocycle satisfies for every nonempty with
Coboundary
A qcochain is called a qcoboundary if it is in the image of and is the set of all qcoboundaries.
For example, a 1cochain is a 1coboundary if there exists a 0cochain 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 {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 wellbehaved 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 AlexanderSpanier cohomology.
For a presheaf on X, let denote its sheafification. Then we have a natural comparison map
from Čech cohomology to sheaf cohomology. If X is paracompact Hausdorff, then is an isomorphism. More generally, is an isomorphism whenever the Čech cohomology of all presheaves on X with zero sheafification vanishes.^{[2]}
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 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 rfold intersections of open subsets inside the ambient topological space are replaced by the rfold 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 quasicoherent 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 quasiprojective over an affine scheme.^{[3]}
The possible difference between Čech cohomology and sheaf cohomology is a motivation for the use of hypercoverings: these are more general objects than the Čech nerve
A hypercovering K_{∗} of X is a certain simplicial object in C, i.e., a collection of objects K_{n} together with boundary and degeneracy maps. Applying a sheaf to K_{∗} yields a simplicial abelian group whose nth cohomology group is denoted . (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.^{[4]}
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
References
Citation footnotes
 ^ Penrose, Roger (1992), "On the Cohomology of Impossible Figures", Leonardo, 25 (3/4): 245–247, doi:10.2307/1575844, JSTOR 1575844, S2CID 125905129. 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
 ^ Brady, Zarathustra. "Notes on sheaf cohomology" (PDF). p. 11. Archived (PDF) from the original on 20220617.
 ^ Milne, James S. (1980), "Section III.2, Theorem 2.17", Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, ISBN 9780691082387, MR 0559531
 ^ Artin, Michael; Mazur, Barry (1969), "Lemma 8.6", Etale homotopy, Lecture Notes in Mathematics, vol. 100, Springer, p. 98, ISBN 9783540361428
General references
 Bott, Raoul; Loring Tu (1982). Differential Forms in Algebraic Topology. Springer. ISBN 0387906134.
 Hatcher, Allen (2002). Algebraic Topology (PDF). Cambridge University Press. ISBN 0521795400.
 Wells, Raymond (1980). "2. Sheaf Theory: Appendix A. Cech Cohomology with Coefficients in a Sheaf". Differential Analysis on Complex Manifolds. Springer. pp. 63–64. doi:10.1007/9781475739466_2. ISBN 9783540904199.