Depth and the local cohomology of FIG-modules

In this paper we describe a machinery for homological calculations of representations of FIG, and use it to develop a local cohomology theory over any commutative Noetherian ring. As an application, we show that the depth introduced by the second author in [16] coincides with a more classical invariant from commutative algebra, and obtain upper bounds of a few important invariants of FIG-modules in terms of torsion degrees of their local cohomology groups.