By the way, regarding the conversation the other day mathoverflow.net/questions/118500/…

It looks like Akhil has been thinking about this for a while

Jacob's answer seems to suggest something I was reluctant to speculate about: Perhaps working with PD E-infty rings, however they are defined, would remove some of the weird features of spectral AG, like the two models of the affine line. Indeed, Jacob says in his answer that working over a field k, forcing both models of A^1 to be equivalent is the passage from E-infty ring spectra over k to SCRs over k

So whatever Akhil is thinking about, maybe the idea is to try to do DAG over a deeper base than spec Z

and then at the risk of speculating really hard, maybe this is where F_1 lives

@TomBachmann the map X -> 0 induces a map Sym(X) -> 1 to the unit, and you can complete with respect to this map. there are a number of references on such completions. it used to be in chapter 4 of DAG XII, but now it looks like it's moved to chapter 7 of SAG. however, a ton of that is a replay of things that are in papers like Greenlees-May's "localization and completion in algebra and topology" or Bousfield's work on completion of spectra

there's also Carlsson and Baker who both did a lot of work on completion for R-modules

I'm ambarassed a bit to ask this but here goes:

What happens if i take some (lubin tate) morava E-theory and try to kill the non-invertible $v_n$'s inside the category of $E_1$-algebras? Do I get a morava K-theory if I forget the algebra structure? Do I get something else? Maybe its not known?

@TylerLawson Ah yes, that's a more hands on way of looking at it. Thanks!

Basically i'm trying to understand whether there is in any sense a canonical map from E-theory to its "residue field" mimicking $\mathbb{Z}_p \to \mathbb{F}_p$

@DenisNardin does this mean your claim was wrong?

This means that I misread the message, and so probably was wrong

Quotienting "as an E_1-ring" is a weird operation

Is it true that K(n) is an E_1-E-algebra?

I guess yes by the abtstract of arxiv.org/pdf/0810.5032.pdf

Yes, it is true. But there are multiple choices about the E_1-algebra structure

uncountably many according to the article I linked Oo

@DenisNardin Could you please give an example illustrating why "quotienting as an E_1-ring is a weird operation"? It would help a lot.

@SaalHardali there isn't a canonical such map (because there isn't a canonical K(n), as an algebra). In fact, this is a big part of the Hopkins-Lurie stuff on the Brauer group: math.harvard.edu/~lurie/papers/Brauer.pdf

but one model for this map is via Thom spectra, which is explained in that paper for example, and from this construction it's evident where at least some of the ambiguity of the quotienting comes from

@DylanWilson Thanks, actually I was kind of embarrased to ask the question precisely because i was aware of HL. Its just i cant shake the question of what happens when one just tries to take the quotient in E_1 algebras. Can one show that it gives the wrong homotopy groups at least? Or maybe even just a basic abstract argument which indicates why this is a wrong question to ask...