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?