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Thanks!
I have added more hyperlinks. And linked to the entry from K-theory.
I am adding various little bits to Morava K-theory, but not done yet.
So I might be wrong here, but my understanding (and maybe I made a mistake when I created this page??) was that Morava K-theory, without any adjectives, has coefficient ring $\mathbb{F}_p[v_n^{\pm 1}]$, where I identify $\mathbb{F}_p$ with $\mathbb{Z}/(p)$. I thought that $\mathbb{Z}_p$ is typically the $p$-adics, and $\mathbb{Z}_{(p)}$ is typically the integers localized at $(p)$.
Anyway, my main point is that Morava K-theory’s coefficients are over $\mathbb{F}_p$.
Yeah in general there seems to be some inconsistency notationally on that page. Perhaps this is my fault. I can’t remember.
Right, I messed it up a bit. Give me a minute to fix…
Okay, I went through it and tried to make the notation for coefficients be both sensible and consistent.
I’d hope to further expand the entry eventually. Besides the abstract characterization currently given, there should also be a concrete construction, etc. I’ll have to see how far I get.
There’s a contradiction between Proposition 2 and Remark 1… Even if $p$ is odd, is it even true that $K(n)$ is $E_\infty$? I thought that was unknown, at least. (Morava E-theory, on the other hand, is $E_\infty$ by the Goerss-Hopkins-Miller theorem.)
Thanks for catching this. I changed in Prop. 2 “$E_\infty$” to “$A_\infty$”.
(Sorry for the glitches. But I am happy about the feedback…)
following up on the issue regarding $A_\infty$/$E_\infty$-structure:
so I gather $K(n)$ becomes $E_\infty$ after all, not over $\mathbb{S}$, but over some $\widehat{E (n)}$.
Similarly Morava $E(n)$ is $E_\infty$ over $B P$.
According to 2.2, 2.3 in
based on
I think Andrew Baker follows the convention by which “(commutative) $R$-ring spectrum” means (commutative) monoid in the homotopy category of $R$-modules. If $K(n)$ really were $E_\infty$ over $\widehat{E(n)}$, then it would also be $E_\infty$ over $\mathbb{S}$, because the $(\infty,1)$-category of $E_\infty$-objects in $R$-modules is equivalent to that of $E_\infty$-ring spectra under $R$. I wonder if $K(n)$ is expected to be $E_\infty$ in some category of pro-module spectra over Morava $E$-theory, being a “residue field” of that derived affine ind-scheme.
oh, okay, darn. Thanks.
The sections on “Universal Characterization” and “Ring Structure” are still messed up.
Lurie does not state in lecture 24 that $K(n)$ is the unique $A_\infty$ ring with such properties 1,2,3. Rather, he proves that any homotopy associative ring spectrum with properties 1,2,3 has underlying spectrum equivalent to $K(n)$.
Robinson (and Baker at $p=2$) proved that $K(n)$ admits an $A_\infty$-ring structure. A remarkable result of Vigleik Angeltveit (“Uniqueness of Morava $K$-theory”) says that if $R$ and $R'$ are $A_\infty$ rings whose underlying spectra admit an equivalence to $K(n)$, then there exists an equivalence $f\colon R\to R'$ as $A_\infty$-rings.
This provides a uniqueness result for $K(n)$ as an $A_\infty$-ring, I guess. The uniqueness statement in the nforum page is still a bit misleading. It would be better to have property 1 say that it is an $A_\infty$-ring $R$ which admits a complex orientation. The statement complex oriented suggests a given choice of ring map $MU\to R$ has been given; but such a choice is not unique, not even up to equivalence of the underlying spectrum of $R$.
Having just written this, I’ve discovered there are many things I’m confused about. For instance, one may instead ask if (P) is true:
(P) Given any two homotopy associative ring spectra $R$ and $R'$ whose underlying spectrum admits an equivalence to $K(n)$, there exists an equivalence $f\colon R\to R'$ of homotopy associative ring spectra.
I think (P) is not true. But it is hard to see how to square that with the results of Robinson, Baker, and Vigleik.
Okay, thanks. If nobody else does, I’ll fix this tomorrow.
have added DOI to:
added pointer to:
Hm, let’s see. This statement originates way back in revision 1, where it seems to be attributed to Ravenel, Douglas: Nilpotence and Periodicity in Stable Homotopy Theory.
Updated part 6 of axiomatic characterization to include the phrase “for n sufficiently large” and remove an unnecessary hypothesis (non-contractibility). This matches the statement of Proposition 1.5.2 in Ravenel’s orange book, “Nilpotence and Periodicity in Stable Homotopy Theory”.
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