Quick Background: The $p$-series of $F$ (where $F$ is a formal group law over a graded ring $R$) will be of the form $[p](x) = px + v\_1x^{p^1} + ... + v\_nx^{p^n} + ...$ ; $(F, R)$ is Landweber-exact if the sequence $(p, v\_1, ..., v\_n)$ is $R$-regular for all $p$ and $n$.


Recall the the Landweber-Ravenel-Stong Construction: $MU^\*(X) \otimes\_{L} R \simeq E^\*(X)$, where $MU^\* \simeq L$ and $R \simeq E^\*(pt)$. 


The Landweber-exact functor theorem states that if $(F, R)$ is Landweber-exact, then $E^\*(X)$ will satisfy the generalized Eilenberg-Steenrod axioms.


An object $M$ in an abelian tensor category $C$ is 'flat' if for all $X \in Obj(C)$ , the functor $X \to X \otimes M$ preserves exact sequences.




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What confuses me is the statement that requiring $L \to R$ to be Landweber-exact is a weaker condition than flat. 


If we are requiring the functor $- \otimes\_L R: MU^\*(X) \to E^\*(X)$ preserves exact sequences, then wouldn't it automatically be flat?


What is an example of a formal group law that is Landweber-exact but not flat?



What is an example of a formal group law that is Landweber-exact but not flat?
Consider the functor that sends $X$ to $MU\_\*(X) \otimes\_{MU\_\*} R$. If $R$ is a flat $MU\_\*$-module then this defines a homology theory. Note that the condition that $R$ is flat over $MU\_\*$ is equivalent to requiring that $\operatorname{Tor}\_1^{MU\_\*}(-,R) = 0$. But we can get away with something weaker; namely we only require that $\operatorname{Tor}\_1^{MU\_\*}(MU\_\*X,R) = 0$ for $X$ a finite complex. It is this requirement that leads (after some work) to Landweber's criterion. 


Since you are interested in elliptic cohomology you can take the original [Landweber-Ravenel-Stong](http://www.math.rochester.edu/people/faculty/doug/mypapers/lrs.pdf) construction (see Section 4), as an example of a cohomology theory arising from a ring that is not flat over $MU\_\*$.


