  In this paper we describe a machinery for homological calculations of
representations of FI_G, 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 coincides with a more classical invariant from
commutative algebra, and obtain upper bounds of a few important invariants of
FI_G-modules in terms of torsion degrees of their local cohomology groups.
