TITLE: Coherent MU_*-Modules
QUESTION [7 upvotes]: It is proven by Thom that for a finite cw-complex $X$, its $MU$-homology, which, in honor of the authors I'm currently reading, I'll denote by $\Omega_\ast^U(X)$, is a coherent module over $\Omega_\ast^U$ if $\Omega_\ast^U(X)$ has projective dimension 0 or 1 over $\Omega_\ast^U$.  It is stated that in a series of lecture notes by Larry Smith that this result can probably be extended to other complexes (and spectra...).  However, these lecture notes are from 1970.  Does anyone know if this result has been fully generalized? I guess I mean, is it known precisely how far this result can be extended?
Thanks!

REPLY [7 votes]: The question is also addressed in Lecture 5 of J.F. Adams ``Lectures on generalized cohomology'' in Springer Lecture Notes in Mathematics Vol 99(1969).  Again ancient, but none the worse for that. 
The paper is reprinted in Volume I of "The selected works of J. Frank Adams''.