Čech-to-derived functor spectral sequence explained
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
be a sheaf on a topological space
X. Choose an open cover
of
X. That is,
is a set of open subsets of
X which together cover
X. Let
denote the presheaf which takes an open set
U to the
qth cohomology of
on
U, that is, to
. For any presheaf
, let
denote the
pth Čech cohomology of
with respect to the cover
. Then the Čech-to-derived functor spectral sequence is:
=\check{H}p(ak{U},l{H}q(X,l{F})) ⇒ Hp+q(X,l{F}).
Properties
If
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 E2-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
is a quasi-coherent sheaf on a
scheme and each element of
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
