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    • CommentRowNumber1.
    • CommentAuthorJon Beardsley
    • CommentTimeMar 6th 2012
    Created a page Morava K-theory . A lot to add. Will fill out later, with better reference list. Please edit!
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 6th 2012

    Thanks!

    I have added more hyperlinks. And linked to the entry from K-theory.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2013

    I am adding various little bits to Morava K-theory, but not done yet.

    • CommentRowNumber4.
    • CommentAuthorJon Beardsley
    • CommentTimeJun 17th 2013
    • (edited Jun 17th 2013)

    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 𝔽 p[v n ±1]\mathbb{F}_p[v_n^{\pm 1}], where I identify 𝔽 p\mathbb{F}_p with /(p)\mathbb{Z}/(p). I thought that p\mathbb{Z}_p is typically the pp-adics, and (p)\mathbb{Z}_{(p)} is typically the integers localized at (p)(p).

    Anyway, my main point is that Morava K-theory’s coefficients are over 𝔽 p\mathbb{F}_p.

    • CommentRowNumber5.
    • CommentAuthorJon Beardsley
    • CommentTimeJun 17th 2013

    Yeah in general there seems to be some inconsistency notationally on that page. Perhaps this is my fault. I can’t remember.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2013

    Right, I messed it up a bit. Give me a minute to fix…

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2013
    • (edited Jun 17th 2013)

    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.

    • CommentRowNumber8.
    • CommentAuthorMarc Hoyois
    • CommentTimeJun 17th 2013

    There’s a contradiction between Proposition 2 and Remark 1… Even if pp is odd, is it even true that K(n)K(n) is E E_\infty? I thought that was unknown, at least. (Morava E-theory, on the other hand, is E E_\infty by the Goerss-Hopkins-Miller theorem.)

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 17th 2013
    • (edited Jun 17th 2013)

    Thanks for catching this. I changed in Prop. 2 “E E_\infty” to “A A_\infty”.

    (Sorry for the glitches. But I am happy about the feedback…)

    • CommentRowNumber10.
    • CommentAuthorstilson
    • CommentTimeSep 17th 2013
    I removed the portion saying 'hence an A_infty ring spectrum' as it contradicts later parts of the entry. In fact, it is known that morava K theory can not be H_infty, one reference is Mark Steinbergers portion of the H_infty ring spectra volume. I will add that later ...
    • CommentRowNumber11.
    • CommentAuthorstilson
    • CommentTimeSep 17th 2013
    Also, references to Vigleik Angeltveit and Alan Robinson's work should be made.
    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2013
    • (edited Nov 15th 2013)

    following up on the issue regarding A A_\infty/E E_\infty-structure:

    so I gather K(n)K(n) becomes E E_\infty after all, not over 𝕊\mathbb{S}, but over some E(n)^\widehat{E (n)}.

    Similarly Morava E(n)E(n) is E E_\infty over BPB P.

    According to 2.2, 2.3 in

    based on

    • Neil Strickland, Products on MUMU-modules, Trans. Amer. Math. Soc. 351 (1999), 2569-2606.
    • CommentRowNumber13.
    • CommentAuthorMarc Hoyois
    • CommentTimeNov 15th 2013

    I think Andrew Baker follows the convention by which “(commutative) RR-ring spectrum” means (commutative) monoid in the homotopy category of RR-modules. If K(n)K(n) really were E E_\infty over E(n)^\widehat{E(n)}, then it would also be E E_\infty over 𝕊\mathbb{S}, because the (,1)(\infty,1)-category of E E_\infty-objects in RR-modules is equivalent to that of E E_\infty-ring spectra under RR. I wonder if K(n)K(n) is expected to be E E_\infty in some category of pro-module spectra over Morava EE-theory, being a “residue field” of that derived affine ind-scheme.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeNov 15th 2013

    oh, okay, darn. Thanks.

    • CommentRowNumber15.
    • CommentAuthorCharles Rezk
    • CommentTimeNov 16th 2013

    The sections on “Universal Characterization” and “Ring Structure” are still messed up.

    Lurie does not state in lecture 24 that K(n)K(n) is the unique A 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)K(n).

    Robinson (and Baker at p=2p=2) proved that K(n)K(n) admits an A A_\infty-ring structure. A remarkable result of Vigleik Angeltveit (“Uniqueness of Morava KK-theory”) says that if RR and RR' are A A_\infty rings whose underlying spectra admit an equivalence to K(n)K(n), then there exists an equivalence f:RRf\colon R\to R' as A A_\infty-rings.

    This provides a uniqueness result for K(n)K(n) as an A 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 A_\infty-ring RR which admits a complex orientation. The statement complex oriented suggests a given choice of ring map MURMU\to R has been given; but such a choice is not unique, not even up to equivalence of the underlying spectrum of RR.


    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 RR and RR' whose underlying spectrum admits an equivalence to K(n)K(n), there exists an equivalence f:RRf\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.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2013

    Okay, thanks. If nobody else does, I’ll fix this tomorrow.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2020

    have added DOI to:

    • Urs Würgler, Morava K-theories: a survey, Algebraic topology Poznan 1989, Lecture Notes in Math. 1474, Berlin: Springer, pp. 111 138 (1991) (doi:10.1007/BFb0084741)

    diff, v36, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2020

    added pointer to:

    diff, v36, current

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