Gross-Hopkins duality and the Gorenstein condition

TL;DR

This paper provides a conceptual interpretation of the near-coincidence of Spanier-Whitehead and Brown-Comenetz dualities in chromatic homotopy theory through the Gorenstein condition for maps of ring spectra, linking duality phenomena to algebraic conditions.

Contribution

It introduces a general notion of Brown-Comenetz dualizing modules for ring spectrum maps and establishes a bijective correspondence with invertible K(n)-local spectra.

Findings

Dualizing modules correspond bijectively to invertible K(n)-local spectra.

The Gorenstein condition explains the relationship between different dualities.

A conceptual framework unifies duality phenomena in chromatic stable homotopy theory.

Abstract

Gross and Hopkins have proved that in chromatic stable homotopy, Spanier-Whitehead duality nearly coincides with Brown-Comenetz duality. Our goal is to give a conceptual interpretation for this phenomenon in terms of the Gorenstein condition for maps of ring spectra in the sense of [Duality in algebra and topology, Adv. Math. 200 (2006), 357--402. arXiv: math.AT/0510247 ]. We describe a general notion of Brown-Comenetz dualizing module for a map of ring spectra and show that in this context such dualizing modules correspond bijectively to invertible K(n)-local spectra.