Local duality for structured ring spectra

TL;DR

This paper develops a general framework for local duality in structured ring spectra, computes local cohomology for finite morphisms, and applies these results to modular representation theory of groups and schemes.

Contribution

It generalizes the local duality theorem to Noetherian $ ext{E}_ olinebreak{}_ olinebreak{}_ olinebreak{}_ olinebreak{}_ olinebreak{}_ olinebreak{}_ olinebreak{}$-ring spectra and connects it to representation theory of groups and schemes.

Findings

Computed local cohomology of dualizing modules for finite morphisms.

Generalized local duality theorem for structured ring spectra.

Applied results to modular representation theory of Lie groups and finite group schemes.

Abstract

We use the abstract framework constructed in our earlier paper to study local duality for Noetherian $\mathbb{E}_{\infty}$-ring spectra. In particular, we compute the local cohomology of relative dualizing modules for finite morphisms of ring spectra, thereby generalizing the local duality theorem of Benson and Greenlees. We then explain how our results apply to the modular representation theory of compact Lie groups and finite group schemes, which recovers the theory previously developed by Benson, Iyengar, Krause, and Pevtsova.