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 ⨁n≥0Ht(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.Comment: Fixed some minor mistakes and expanded the section on configuration
space