Čech-to-derived functor spectral sequence
In algebraic topology, a branch of mathematics, the Čech-to-derived functor spectral sequence is a spectral sequence that relates Čech cohomology of a sheaf and sheaf cohomology. Definition. Let formula_1 be a sheaf on a topological space "X". Choose an open cover formula_2 of "X". That is, formula_2 is a set of open subsets of "X" which together cover "X". Let formula_4 denote the presheaf which takes an open set "U" to the "q"th cohomology of formula_1 on "U", that is, to formula_6. For any presheaf formula_7, let formula_8 denote the "p"th Čech  cohomology of formula_7 with respect to the cover formula_2. Then the Čech-to-derived functor spectral sequence is: Properties. If formula_2 consists of only two open sets, then this spectral sequence degenerates to the Mayer–Vietoris sequence. See Spectral sequence#Long exact sequences. If for all finite intersections of a covering the cohomology vanishes, the "E"-term degenerates and the edge morphisms yield an isomorphism of Čech cohomology for this covering to sheaf cohomology. This provides a method of computing sheaf cohomology using Čech cohomology. For instance, this happens if formula_1 is a quasi-coherent sheaf on a scheme and each element  of formula_2 is an open affine subscheme such that all finite intersections are again affine (e.g. if the scheme is separated). This can be used to compute the cohomology of line bundles on projective space.