Local Gorenstein duality in chromatic group cohomology

TL;DR

This paper investigates local Gorenstein duality in the context of chromatic group cohomology, demonstrating its systematic occurrence across various complex orientable ring spectra and establishing a descent method for broader applicability.

Contribution

It introduces a systematic analysis of local Gorenstein duality for cochain spectra on classifying spaces and proves a descent result to extend the range of examples.

Findings

Local Gorenstein duality holds for Lubin-Tate theories, topological K-theory, and topological modular forms.

A descent theorem enables access to additional examples of local Gorenstein duality.

The results unify various instances of duality in chromatic group cohomology.

Abstract

We consider local Gorenstein duality for cochain spectra $C^*(BG;R)$ on the classifying spaces of compact Lie groups $G$ over complex orientable ring spectra $R$. We show that it holds systematically for a large array of examples of ring spectra $R$, including Lubin-Tate theories, topological $K$-theory, and various forms of topological modular forms. We also prove a descent result for local Gorenstein duality which allows us to access further examples.