added pointer to:
have added DOI to:
Okay, thanks. If nobody else does, I’ll fix this tomorrow.
]]>The sections on “Universal Characterization” and “Ring Structure” are still messed up.
Lurie does not state in lecture 24 that is the unique 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 .
Robinson (and Baker at ) proved that admits an -ring structure. A remarkable result of Vigleik Angeltveit (“Uniqueness of Morava -theory”) says that if and are rings whose underlying spectra admit an equivalence to , then there exists an equivalence as -rings.
This provides a uniqueness result for as an -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 -ring which admits a complex orientation. The statement complex oriented suggests a given choice of ring map has been given; but such a choice is not unique, not even up to equivalence of the underlying spectrum of .
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 and whose underlying spectrum admits an equivalence to , there exists an equivalence 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.
]]>oh, okay, darn. Thanks.
]]>I think Andrew Baker follows the convention by which “(commutative) -ring spectrum” means (commutative) monoid in the homotopy category of -modules. If really were over , then it would also be over , because the -category of -objects in -modules is equivalent to that of -ring spectra under . I wonder if is expected to be in some category of pro-module spectra over Morava -theory, being a “residue field” of that derived affine ind-scheme.
]]>following up on the issue regarding /-structure:
so I gather becomes after all, not over , but over some .
Similarly Morava is over .
According to 2.2, 2.3 in
based on
Thanks for catching this. I changed in Prop. 2 “” to “”.
(Sorry for the glitches. But I am happy about the feedback…)
]]>There’s a contradiction between Proposition 2 and Remark 1… Even if is odd, is it even true that is ? I thought that was unknown, at least. (Morava E-theory, on the other hand, is by the Goerss-Hopkins-Miller theorem.)
]]>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.
]]>Right, I messed it up a bit. Give me a minute to fix…
]]>Yeah in general there seems to be some inconsistency notationally on that page. Perhaps this is my fault. I can’t remember.
]]>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 , where I identify with . I thought that is typically the -adics, and is typically the integers localized at .
Anyway, my main point is that Morava K-theory’s coefficients are over .
]]>I am adding various little bits to Morava K-theory, but not done yet.
]]>Thanks!
I have added more hyperlinks. And linked to the entry from K-theory.
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