Its a standard fact that over $\mathbb{Q}$ there's a canonical isomorphism $MU_* \otimes \mathbb{Q} \cong \mathbb{Q}[\mathbb{CP}^1,]\mathbb{CP}^2,\dots]$. Is it true that denoting $z_n = \frac{\mathbb{CP}^n}{n+1}$ there's a canonical isomorphism $MU_* \cong \mathbb{Z}[z_1,z_2,\dots]$?
@SaalHardali Unfortunately not true. The elements CP^n/(n+1) are actually the coefficients of the universal logarithm for MU_*, so those elements can't all actually be in the Lazard ring or every formal group law would have a log. In particular CP^1 is a generator for the Lazard ring in degree two and CP^1/2 isn't in there.
(also, if CP^1 was bordant to 2M for some M, then the first Chern number of M would have to be 1/2, that would be bad)
@TylerLawson Actually this is how I got here in the first place. I'm trying to search for an explicit formula for the universal formal group whose coefficients can be identified with some canonical bordism classes genetrating the bordism ring. My rationale was that after tensoring with $\mathbb{Q}$ there's an explicit fotmula for the logarithm.
so maybe one can recover the universal FGL from it
Me and my friend did some computations and it became evident this doesn't work unless the CP^n are divisible by n+1 which then implies that all FGL's have a logarihm so its kind of a hyperbole
Are there explicit bordiam classes generating the Lazard ring as a polynomial algebra?
@SaalHardali you might be interested in Section V.10 of lazard's commutative formal groups, where he describes a method that comes very close to naming a canonical algebraic presentation of the lazard ring. no bordism classes, but even the algebraic question is interesting
@EricPeterson Actually the first motivation was to have a formula for the universal form group law so that I could see explicitly what happens to the FGL when one passes from $MU$ to $BP$ in particular find a way to express $v_n$'s in terms of the universal formula.
Then I realized one needs first canonical generators in order to state this. Then I thought that asking for canonical generators in the Lazard ring was too much but that perhaps Quillen's theorem might provide them.
If {C_i} is a filtered system of sites with colimit C, then the topos of set-valued sheaves on C is the limit of the topoi on the C_i. Is the same true for the oo-toposes?
@JonathanBeardsley in the case "O-" = "symmetric", only lax. (i like to think about the 'free vector space on a set' functor when i forget which way things go or something.)
I feel like this is probably somewhere in Lurie? I'm actually only interested in the case of the Wirthmuller context going between spaces parameterized by $O$-algebras (really just $E_n$-algebra actually!). In that case, the pullback functor $f^\ast$ is $O$-monoidal, but I can't find anything in the literature about its right adjoint $f_\ast$.
