  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.
