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?