A homotopyish Landweber exact functor theorem Let $M$ be a $\pi_*(MU)$-module. The Landweber exact functor theorem gives conditions for the functor that sends a space $X$ to $ MU(X) \otimes_{\pi_*(MU)} M$ to define a homology theory on spaces, which thus comes from a spectrum. 
It'd be nice, though, if one could construct the spectrum directly, instead of going through the homology theory. For instance, it would be nice if one could construct an actual $MU$-module (possibly under further hypotheses) or an $MU$-algebra when $M$ is an algebra. Is there another version of the exact functor theorem that lets one do this? 
 A: I'm not sure that this is exactly what you are looking for, but I looked a bit
at the Landweber exact functor theorem in the context of $MU$-modules
at the end of a very short paper: Idempotents and Landweber exactness in 
brave new algebra. Homology, homotopy, and applications 3(2001), 355--359.
Theorem 8 there reads: If $M_*$ is a Landweber exact $MU_*$-module, then 
there is an $MU$-module $M$ such that $\pi_*(M) = M_*$ and, for any finite 
cell $MU$-module $X$, $\pi_*(X)\otimes_{MU_*} M_* \cong \pi_*(X\wedge_{MU} M)$.
A: Here are three methods that I know:


*

*In the case $M_*=(MU_*/I)[S^{-1}]$ (where $I$ is generated by a regular sequence) there is a more direct construction by reducing to the cases $M_*=MU_*/a$ and $M_*=MU_*[a^{-1}]$.  My paper 'Products on MU-modules' is probably the sharpest version, but there are many earlier versions in a similar spirit.

*In the case $M_*=MU_*[x_1,\dotsc,x_r]$ with $|x_i|=0$ you can use $MU\wedge\Sigma^\infty_+\mathbb{N}^r$ (and this has an $E_\infty$ structure).

*In the case $M_*=MU_*[n^{-1}]$ (for some $n\in MU_0=\mathbb{Z}$) you can note that there are natural $E_\infty$ maps
$$ MU\xleftarrow{f}\Sigma^\infty_+DS^0\xrightarrow{}\Sigma^\infty_+QS^0,$$
where $f$ has degree $n$ on the bottom cell.  The smash product 
$$ MU\wedge_{\Sigma^\infty_+DS^0}\Sigma^\infty_+QS^0$$
then has the required property.
There are some fairly obvious ways to combine these methods and generalise them slightly.
Under the general conditions of the Landweber theorem, I know of several people including myself who have looked quite hard for a more direct construction, but without success.
A: Akhil, short though that paper is, I claim that the proof there is as
complete as it needs to be. 
