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.