  In this work we study a kind of coherence condition on FI_G-modules, which
generalizes the usual notion of finite generation. We prove that a module is
coherent, in the appropriate sense, if and only if its generators, as well as
its torsion, appears in only finitely many degrees. Using this technical
result, we prove that the category of coherent FI_G-modules is abelian,
independent of any assumptions on the group G, or the coefficient ring k.
Following this, we consider applications towards the local cohomology theory of
FI_G-modules, introduced by Li and the author in previous work.
