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.
The idea of Čech cohomology is that, for an open cover
consisting of sufficiently small open sets, the resulting simplicial complex
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.
is an ordered collection of q+1 sets chosen from
This intersection is called the support of σ and is denoted |σ|.
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
is a map which associates with each q-simplex σ an element of
The coboundary operator is analogous to the exterior derivative of De Rham cohomology, so it sometimes called the differential of the cochain complex.
satisfying a compatibility relation on every intersecting
Thus the qth Čech cohomology is given by The Čech cohomology of X is defined by considering refinements of open covers.
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 coefficients in a fixed abelian group A, denoted
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 {ρi} such that each support
If X is homotopy equivalent to a CW complex, then the Čech cohomology
is also naturally isomorphic to the de Rham cohomology; the article on de Rham cohomology provides a brief review of this isomorphism.
For example if X is the closed topologist's sine curve, then
If X is compact Hausdorff, then Čech cohomology (with coefficients in a discrete group) is isomorphic to Alexander-Spanier cohomology.
is an isomorphism whenever the Čech cohomology of all presheaves on X with zero sheafification vanishes.
[2] Čech cohomology can be defined more generally for objects in a site C endowed with a topology.
is defined as where the colimit runs over all coverings (with respect to the chosen topology) of X.
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.
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.
[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 Kn together with boundary and degeneracy maps.
Then, it can be shown that there is a canonical isomorphism where the colimit now runs over all hypercoverings.
[4] The most basic example of Čech cohomology is given by the case where the presheaf
; the fourth gives Such a function is fully determined by the values of
Using the cover we have the following modules from the cotangent sheaf If we take the conventions that
