  We introduce a formalism to produce several families of spectral sequences
involving the derived functors of the limit and colimit functors over a finite
partially ordered set. The first type of spectral sequences involves the left
derived functors of the colimit of the direct system that we obtain applying a
family of functors to a single module. For the second type we follow a
completely different strategy as we start with the inverse system that we
obtain by applying a covariant functor to an inverse system. The spectral
sequences involve the right derived functors of the corresponding limit. We
also have a version for contravariant functors. In all the introduced spectral
sequences we provide sufficient conditions to ensure their degeneration at
their second page. As a consequence we obtain some decomposition theorems that
greatly generalize the well-known decomposition formula for local cohomology
modules given by Hochster.
