Periodicity in the cohomology of symmetric groups via divided powers

A famous theorem of Nakaoka asserts that the cohomology of the symmetric group stabilizes. The first author generalized this theorem to non-trivial coefficient systems, in the form of FI-modules over a field, though one now obtains periodicity of the cohomology instead of stability. In this paper, we further refine these results. Our main theorem states that if M is a finitely generated FI-module over a noetherian ring k then circle plus (n >= 0) H-t(Sn, Mn) admits the structure of a D-module, where D is the divided power algebra over k in a single variable, and moreover, this D-module is 'nearly' finitely presented. This immediately recovers the periodicity result when k is a field, but also shows, for example, how the torsion varies with n when k = Z. Using the theory of connections on D-modules, we establish sharp bounds on the period in the case where k is a field. We apply our theory to obtain results on the modular cohomology of Specht modules and the integral cohomology of unordered configuration spaces of manifolds.