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. 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 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 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. Applying a sheaf to K∗ yields a simplicial abelian group whose n-th 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. 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 General references Cohomology theories