  Homology decomposition techniques are a powerful tool used in the analysis of
the homotopy theory of (classifying) spaces. The associated Bousfield-Kan
spectral sequences involve higher derived limits of the inverse limit functor.
We study the impact of depth conditions on the vanishing of these higher limits
and apply our theory in several cases. We will show that the depth of
Stanley-Reisner algebras can be characterized in combinatorial terms of the
underlying simplicial complexes, the depth of group cohomology in terms of
depth of group cohomology of centralizers of elementary abelian subgroups, and
the depth of polynomial invariants in terms of depth of polynomial invariants
of point-wise stabilizer subgroups. The latter two applications follow from the
analysis of an algebraic version of centralizer decompositions in terms of
Lannes' $T$-functor.
