TITLE: What is an example of a formal group law that is Landweber-exact but not flat?
QUESTION [7 upvotes]: 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.

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?

REPLY [12 votes]: 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 construction (see Section 4), as an example of a cohomology theory arising from a ring that is not flat over $MU_*$.