28 research outputs found
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 -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 is
a finitely generated -module over a noetherian ring
then admits the structure of a
-module, where is the divided power algebra over
in a single variable, and moreover, this -module is
"nearly" finitely presented. This immediately recovers the periodicity result
when is a field, but also shows, for example, how the torsion
varies with when . Using the theory of connections
on -modules, we establish sharp bounds on the period in the case
where 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