Periodicity in the cohomology of symmetric groups via divided powers

Abstract

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.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143618/1/plms12107.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/143618/2/plms12107_am.pd