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There are some criteria which tell us when a spectrum $X$ is chromatically complete (it's the homotopy limit of its chromatic tower):

It has to be p-local and finite, according to the chromatic convergence theorem, or it has to have finite projective BP dimension [Barthel, 2016].

Do we have a theorem or something similar which tells when a spectrum $X$ is NOT chromatically complete?

In other words, i want to start the analysis of a certain spectrum and i would like to show if it is chromatically complete or not. How can i do that?

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    $\begingroup$ You could try to show that it does not agree with its harmonic localization; or you could try to argue like in Theorem 5.7 of Barthel's paper. If these do not work, please tell us something more about your spectrum. $\endgroup$ – Lennart Meier Nov 25 '17 at 21:35

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