ÄŒech-to-derived Functor Spectral Sequence

In algebraic topology, a branch of mathematics, the ÄŒech-to-derived functor spectral sequence is a spectral sequence that relates ÄŒech cohomology of a sheaf and sheaf cohomology.

Definition

Let $\mathcal$ be a sheaf on a topological space ''X''. Choose an open cover $\mathfrak$ of ''X''. That is, $\mathfrak$ is a set of open subsets of ''X'' which together cover ''X''. Let $\mathcal^q(X, \mathcal)$ denote the presheaf which takes an open set ''U'' to the ''q''th cohomology of $\mathcal$ on ''U'', that is, to $H^q(U, \mathcal)$. For any presheaf $\mathcal$, let $\check^p(\mathfrak, \mathcal)$ denote the ''p''th ÄŒech cohomology of $\mathcal$ with respect to the cover $\mathfrak$. Then the ÄŒech-to-derived functor spectral sequence is:
:$E^_2 = \check^p(\mathfrak, \mathcal^q(X, \mathcal)) \Rightarrow H^(X, \mathcal).$

Properties

If $\mathfrak$ 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''_''2''-term degenerates and the edge morphisms yield an isomorphism of ÄŒech cohomology for this covering to sheaf cohomology. This provides a method of computing sheaf cohomology using ÄŒech cohomology. For instance, this happens if $\mathcal$ is a quasi-coherent sheaf on a scheme and each element of $\mathfrak$ 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.

See also

* Leray's theorem

Notes

References

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{{DEFAULTSORT:Cech-To-Derived funcTor Spectral Sequence
Spectral sequences