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 homotopyish Landweber exact functor theorem
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.


