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On the degree-wise coherence of FI_G-modules

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.