$FGL(R)$ will be the category of formal group laws over the ring $R$. If $(E,x_E)$ is an oriented spectrum (i.e. $x_E \in \tilde{E}^2(\mathbb{P}^{\infty})$ is sent to the unit $1 \in \tilde{E}^2(\mathbb{P}^1)$ by the pullback of the inclusion $i: \mathbb{P}^1 \hookrightarrow \mathbb{P}^\infty$), then the pullback of the multiplication $\mu$ as H-space of $\mathbb{P}^\infty$ define an element in $FGL(E^*(*))$ in the following way:

$ \mu^*:E^*(*)[[x_E]] \cong E^*(\mathbb{P}^\infty) \to E^*(\mathbb{P}^\infty \times \mathbb{P}^\infty) \cong E^*(*)[[x_1,x_2]] $

where $x_i : = p_i^*(x_E)$ and the isomorphism are given by the collapsing of the Atiyah Hirzebruch spectral sequence. The fgl associated to $(E,x_E)$ is by definition $\mu^*(x_E) \in FGL(E^*(*))$. The Landweber exact theorem 'reverse' this contruction: let $f \in FGL(R)$ be a formal group law over $R$, then by certein hypothesis on $f$ the association

$ X \to MU^\bullet(X)\otimes_{\pi_*(MU)} R$

is a cohomology theory on CW-complexes ($MU$ is the complex cobordism spectrum and the $\pi_*(MU)$-module structure on $R$ is given by the classification map $\pi_*(MU) \to R$, since by Quillen's theorem the fgl of the complex cobordism is the universal one). This correspondence given by the Landweber's theorem seems to be only objectwise.


Given a morphism of formal group laws $\varphi$ over $R$, which is a power series in $R[[x]]$, is there a way to produce a natural transformation of the related cohomology theories induced by $\varphi$? Is there any reference in the literature where i could find an answer?

  • $\begingroup$ Why does the correspondence seems only objectwise? A map of fgls is the same thing as a map of $\pi_*MU$-algebras $\endgroup$ – Denis Nardin Jun 4 '18 at 19:41

Yes, if $E$ and $E'$ are Landweber exact, then ring maps $E\to E'$ biject with isomorphisms of the associated formal groups, suitably interpreted. One version of this (for the context where $E$ and $E'$ are $2$-periodic) is explained in Proposition 8.43 of this Memoir. There is also a rather terse account in Proposition 11 of this lecture of Lurie.

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