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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?

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    $\begingroup$ Landweber exact means tensoring with it preserves exactness for comodules over the Hopf algebroid $(MU_*,MU_*MU)$(Which is automatically satisfied when it is $MU_*(X)$ for some spectrum $X$), which is a subclass of modules over the Lazard's ring. So Landweber-exact is weaker than flat. But I could not come up with any quick example in my mind. $\endgroup$
    – Mingcong Zeng
    Commented Apr 10, 2015 at 5:39
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    $\begingroup$ $\Bbb Q$ is Landweber flat, but certainly not flat over $L$. As Mingcong and Drew point out, what Landweber actually proved was that any $MU_* MU$-comodule is built out of a very restricted family of $L$-modules, and so you only need to test flatness against those basic building blocks rather than a general $L$-module. $\endgroup$
    – Tyler Lawson
    Commented Apr 10, 2015 at 15:07
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    $\begingroup$ If $R$ is flat over $MU_*$, then $R$ has to be $MU_*$-torsion free and thus contains a copy $MU_*\cdot 1$ of $MU_*$. This cannot be true for any ring $R$ of finite dimension. So virtually all examples you can think of are not flat over $MU_*$. E.g. rational homology, K-theory, elliptic homology, Johnson--Wilson theories $E(n)$... $\endgroup$
    – Lennart Meier
    Commented Apr 10, 2015 at 16:31

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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 $MU_*$.

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