Homology
decomposition techniques are a powerful tool used in the analysis of
the homotopy theory of (classifying) spaces. Cohomological calculations
involve higher derived limits of the inverse limit functor. And the
best one can hope for is that most of these higher limits will vanish.
We study the impact of depth conditions for this question. This sounds
quite technical but allows applications in several different areas:
Stanley-Reisner algebras and combinatorics of simplicial complexes,
Group Cohomology, Invariant Theory and Homological Algebra over The
Stenrod algebra. Although the motivation for this work comes from
homotopy theory and geometry, most the theory can be formulated in
algebraic terms. In this talk I will emphasis the algebraic aspect of
the work as well as the applications.