@JonathanBeardsley why should this mean that MU controls derived formal geometry?

@skd because $MU$ (and $BP$) effectively induce the stratification of $\pi_\ast(S)$ (via the ANSS) that looks the moduli stack of formal groups. also, $MU_\ast$ is the Lazard ring

@ArunDebray i might be mistaken, but i'm pretty sure the Kunneth theorem holds whenever you've got field coefficients

in particular there's a Kunneth spectral sequence, and I believe it automatically collapses

@JonathanBeardsley is this stratification supposed to be analogous to the filtration on Spf R coming from powers of the ideal of definition?

@skd i don't think so. it's more just a sort of thing that naturally arises when you start considering this invariant of formal groups called "height." you find that the moduli stack of formal groups has these open substacks of "formal groups of height less than or equal to n"

@skd one thing to notice is that i'm specifically talking about formal groups, not formal schemes, i.e. group objects IN formal schemes. so there's a bit of a leap there.

@JonathanBeardsley I think you also need local finiteness (being finite dimensional in each degree)

sorry, i'm still confused. are you saying that the "correct" notion of derived formal scheme would be one for which group objects in that category are parametrized by a derived lift of M_fg?

@skd perhaps!

@JamesCameron you're probably right!

@skd my claim is something like if you got the right definition of spectral formal scheme and asked about group objects therein then you would recover MU

but this might be impossible and or nonsense!

got it

so is the moduli stack of derived formal groups supposed to be the same as the derived moduli stack of formal groups?

@skd what's a derived formal group? what's the derived moduli stack of formal groups?

sorry -- by a "derived formal group" i meant a group object in derived formal schemes

whatever the correct notion of a derived formal scheme is

So, I'm still not sure what the derived moduli stack of formal groups is, but I think my answer to your question is no.

also: i think that the prototypical example of a derived formal scheme should be Spf E, a derived lift of the Lubin-Tate space, which should have the property that QCoh(Spf E) = K(n)-local E-modules

@JonathanBeardsley i don't know what the derived moduli of formal groups is, either. i'd asked about this before in this chat, and it led to some interesting comments

Yeah I mean, I dunno. Honestly I don't think I'm smart enough to really know the right way to approach this type of stuff. I'm also kind of losing faith that it's worth a lot of energy.

it'd probably be worth somebody sitting down and figuring out what formal schemes should be for cDGAs or something first.

@SaulGlasman Yes, sure. But I was asking if anyone have already developed this. Sketches can codify theories with partial operations, so one can codify groupoids and fields for instance, therefore, in particular it would give another model of \infty-groupoid. It will probably looks like Grothendieck's model, though.

@JonathanBeardsley thanks, I can try running the Künneth spectral sequence

@ArunDebray The Kunneth spectral sequence based on homology with F_2-coeffs collapses because only Tor^0 is non-trivial (since it's Tor over F_2).

oh right, of course

I think that H(X) \otimes H(Y) has the diagonal A-module structure, but I don't know of a reference for that!

it should come from the Cartan formula, right? For example, if a is in H^*(X; F_2) and b is in H^*(Y; F_2), then in H^*(X smash Y; F_2), Sq^1(x ⊗ y) = Sq^1(x) ⊗ y + x ⊗ Sq^1(y)

is that what you meant by the diagonal action? (unsure whether it's that or Sq^1(x ⊗ y) = Sq^1(x) ⊗ Sq^1(y))

@YuriSulyma It sucks over finite characteristics. There's no equivalence with simplicial rings and strict stuff. Furthermore, even in char 0 , the relative affine line (spec of the free E_\infty algebra with one generator) is not flat anymore and also there's no SL_2 over the sphere spectrum. On the other side there's a lower basis , which looks like F_1. Take a look at Lurie's thesis

@ArunDebray Exactly! By diagonal action I meant via the diagonal of A as a Hopf algebra (i.e., its coproduct), which can be read from the Cartan formula

@Bogdan great, thanks!

@ArunDebray anytime

@ArunDebray Even if we didn't know the other one, this formula won't work because the degrees are off.

re: derived formal groups: i think it's unreasonable to expect MU to play the role of a moduli of derived formal groups in the generality y'all intend. the moduli it represents (in the homotopy category) is really different from the one you're used to: the algebroid (MU_, MU_ MU) tracks all (smooth, 1-dim'l) formal groups you can ever find on your favorite graded ring R_, but the algebroid (MU, MU ^ MU) only tracks coordinates on a single *fixed formal group Spf R^* CP^infty

if your definition of 'derived formal group' is wider than 'CP^'infty', MU won't directly know about it

there is a similar classical phenomenon that morava is super fond of: lubin-tate explicit CFT associates to a local number field L a particular (connected) p-divisible group. you can ofc find other formal groups over L, but this is the relevant one that governs CFT

i think a really responsible thing to ask yourself here is: how many other examples do you even want to capture with a definition of 'derived formal group'? and how many of those are more than one-step-removed variations on CP^infty?

i think there might be some out there as variations on the above analogy: even if L has only one formal group that's "relevant", something like k((t)) has a lot of relevant formal groups owing to the large number of abelian varieties over k, and maybe someday homotopy theory will have enough analogues of equi-characteristic objects that you'll find a rich & cool generic theory of derived formal groups lying around

you can imagine other similar fountainheads, but i think we are not there yet, and phenomena should come before theory here

(and, for what it's worth, i think SAG Section 8 is not really after any of this; it sounds like it's trying to recover facts in the neighborhood of: passing to a formal thickening of a closed object in a noetherian parent gives a flat map)

anyway: obviously just my opinion, happy to change it if yall think otherwise

@SeanTilson oh right, thank you!

@EricPeterson i also think that asking for MU to be the analogue of the Lazard ring for "derived formal groups" is too much to ask for

i was wondering, though, if you knew of a notion of derived formal group that's discussed somewhere

i've seen a few places on the internet where people toss that word around, and i don't know where to find anything about this

is this related to the derived p-divisible group story that lurie writes about in the survey paper?

no, i don't know one. i really think if you wanted to have (an exciting) one you'd have to have exciting examples, and i don't even know those

my understanding of the oriented p-divisible group story is that it's part of capturing what extra structure there is in E_infty MU-orientations. there we do have phenomenology to consider: for instance, an E_infty orientation of Morava E-theory is at least the data of an H_infty orientation, and that tracks the finite subgroup structure of the formal group associated via CP^infty to E-theory, a first step of which is carving the thing up into its finite p-power-torsion subgroups

at the very least, p-divisible groups seem to show up in basically the only tool we have for saying anything about E-infinity orientations, I guess for the reason Eric said

Can anyone tell me an interesting topic for real-world applications of integrals? I made an algorithm that solves integrals and would like to apply to some real theme

@DanLucioPrada this room is unlikely to be helpful