  FI-modules were introduced by the first three authors in [CEF] to encode
sequences of representations of symmetric groups. Over a field of
characteristic 0, finite generation of an FI-module implies representation
stability for the corresponding sequence of S_n-representations. In this paper
we prove the Noetherian property for FI-modules over arbitrary Noetherian
rings: any sub-FI-module of a finitely generated FI-module is finitely
generated. This lets us extend many of the results of [CEF] to representations
in positive characteristic, and even to integral coefficients. We focus on
three major applications of the main theorem: on the integral and mod p
cohomology of configuration spaces; on diagonal coinvariant algebras in
positive characteristic; and on an integral version of Putman's central
stability for homology of congruence subgroups.
