@JonathanBeardsley yeah, it's basically the 1-generator case of Nakayama's lemma

@EricPeterson So... does the E-infty ring MU classify derived formal groups? And if not, then what does it classify?

Eric was definitely indicating that MU as a commutative ring does not classify derived formal groups, if such a notion even makes sense. as for what it does classify, well, computing spaces of maps of commutative rings isn't exactly easy

also, there are lots of maps MU_* -> E_* which cannot lift to a map of commutative rings (in spectra) even if the right hand side admits such a structure

@AntonFetisov No one really knows what MU classifies in a derived sense

@AntonFetisov they're right: no one knows. you can reformulate what peter said like this: if there exists a map MU –> E of commutative rings at all, then Spf E^* CP^∞ gives a formal group, and then such maps furthermore biject with coordinates on it

the quantifiers here are really different from what happens in algebra: for any ring R and any formal group G / R and any coordinate x on G, there's a map f: MU_* –> R such that (G, x) = f^* (G_univ, x_univ) arises as the base-change of a fixed (G_univ, x_univ) over MU_*

in the topological setting there is a fixed object G = Spf E^* CP^∞, whose presentations (if it admits presentation as a formal group at all) arise as base-change from Spf MU^* CP^∞

so if your proposed definition of "derived formal group" admits more than one per choise of base ring spectrum, then it's not what's captured by MU

adding "E_∞" only makes the situation more confusing still

idk how appropriate this analogy truly is, but in explicit class field theory, one attaches to a local number ring a particular formal group, its Lubin-Tate group, and extracts information about the number ring through that

the situation in topology seems to be rather near to that