@fpqc the general MO chat is here:

Unfortunately it's significantly less active than this room.

Happy Independence Day

(at least in the United States)

Haha finally I can put a face to the name @TomerSchlank

I guess my secret identity has been revealed. It is indeed I, Terribleblackboardworkman.

Haha, aka the Ultrarabbi.

btw, I week after transchromatic we had a 3 days seminar on the proof of the nilpotence theorem and my talk was strongly influenced by the one you gave in Regensburg. It is now the way I thnk about the X(i)... (The students seemed happy with it too)

Cool! I think that's maybe the right way to think about them, haha.

Are you familiar with Lazard's proof that the Lazard ring is polynomial? It's really suggestive about how to think of the X(i)

Is it the proof using to comparison map to Z[b_1,...] by taking a FGL with logarithm? or is it different?

So if I recall correctly that's how Hopkins does it in COCTALOS for instance, but I think Lazard builds it up level by level (i.e. n-bud to n+1-bud) and basically shows that the obstructions to deforming a commutative 1D n-bud up to an n+1-bud are trivial.

Maybe Lazard also does a comparison at some point using a logarithm. To be honest I haven't looked at his proof in a long time.

It's this one: archive.numdam.org/article/BSMF_1955__83__251_0.pdf

At some point I was trying to prove specific things about n-buds and was staring at that paper (as well as, incidentally, the original Lubin-Tate paper, in which they do something REALLY similar).

Another interesting theorem from that paper is that if a ring doesn't contain any nilpotent elements then every formal group law over it is commutative.

$E(n)$ is the Johnson-Wilson theory. Any element $\lambda\in \mathbb{Z}_{(p)}^\times$ gives an automorphism $[\lambda]$ on the formal group of $E(n)$. Thus $\mathbb{Z}_{(p)}^\times$ acts on $E(n)$ up to homotopy.

Is it known that this action can be rigidified?

I don't think anything like that is known before completion. Is there a known way to do this without obstruction theory? Because I don't think we even know how to make E(n) A_infty by obstruction theory before completion.

I was only wondering about the action on the spectrum, not about any ring structure. I agree having a ring structure would make it easier.

You can also ask for an action of $\mathbb{Z}_p^\times$ on $E(n)_p$, the $p$-completion of $E(n)$.

At $p=2$, you'd want the $\{\pm1\}\subset \mathbb{Z}_{(2)}^\times$ action to pick of the real Johnson-Wilson spectrum of Hu-Kriz.

For comparison, $TMF_{(p)}$ (periodic tmf localized at $p$) does admit a $\mathbb{Z}_{(p)}^\times$-action (through $E_\infty$-maps), by Lurie-theory.

I realize that's what you're asking, I was just saying how the A-infty route isn't known to work (to the best of my knowledge). The only other way I know of (other than magically building the action directly) would be like Cooke's obstruction theory where the obstructions live in like H^*(Z_p^{\times}, \pi_*F(E(n), E(n)))... I dunno how computable that is...

(especially before completion, given how complicated [K, K] is already)

Dylan do you have some odd primary version of BP by now that can build the odd primary version of Real Johnson-Wilson theory?

workin on it

Also- nice paper!

thanks lol. Maybe I can remark that you can make E(n) A_infinity if you are willing to work with a 2-periodic (but uncompleted) version