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There is the Chromatic Convergence Theorem by Hopkins and Ravanel which states that the homotopy inverse limit of the chromatic tower of a finite spectra $X$ is $X$.

Let's call any spectra with this property chromatically complete

There is an article from T. Barthel (Chromatic Completion) which proves a slightly stronger result: to make the tower converge, the spectra $X$ does not have to be finite, but has to have finite projective $BP$-dimension.

The same article also concludes saying that they do not know whether all suspension spectra are chromatically complete and that they believe that the collection of chromatically complete spectra is not closed under infinite products.

So here are my two questions:

Since this article is more than two years old, have we made some progresses about proving or not those two properties?

Besides finite spectra and spectra with finite projective $BP$-dimension, do we know other kinds of spectra that are chromatically complete?

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    $\begingroup$ For $n,k>0$ we can choose a finite suspension spectrum $X_{nk}$ with $BP_*(X_{nk})=\Sigma^dBP_*/(v_0^{i_0},\dotsc,v_{n-1}^{i_{n-1}})$ where $d\geq n+k$ and $i_t\geq k$ for all $t$. Put $X=\bigvee_{n,k}X_{nk}$, which is again a suspension spectrum. The condition on $d$ means that $X=\prod_{n,k}X_{nk}$. This is quite similar to the main counterexample of Bartels and my guess is that it is not chromatically complete. At any rate, it is certainly a key example that one would want to analyse. It might be enough to consider $\bigvee_nX_{n1}$ instead. $\endgroup$ – Neil Strickland Oct 10 '17 at 15:50
  • $\begingroup$ Thank you for the answer, one question: Why can we tell that $X_{nk}$ is a suspension spectrum? $\endgroup$ – Alfred Nov 19 '17 at 17:34

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