Morava modules and Brown-Comenetz duality

Fix a prime number p, and let X be a finite spectrum whose (n - 1)st Morava K-theory is trivial but whose nth Morava K-theory is nontrivial, n > 0. We prove, following a method outlined to us by M. J. Hopkins, that, if 2p > n² + n + 2, the Morava module of the Brown-Comenetz dual of the E(n)_*-localization of X is isomorphic to a suspension of the Pontryagin dual of the Morava module of X. To complete this proof, we found it necessary to develop a more canonical construction of certain modified Adams spectral sequences; this construction should be of independent interest.