  We construct a spectral sequence associated to a stratified space, which
computes the compactly supported cohomology groups of an open stratum in terms
of the compactly supported cohomology groups of closed strata and the reduced
cohomology groups of the poset of strata. Several familiar spectral sequences
arise as special cases. The construction is sheaf-theoretic and works both for
topological spaces and for the \'etale cohomology of algebraic varieties. As an
application we prove a very general representation stability theorem for
configuration spaces of points.
