The localisation L^f_n is compactly central, and the compactly central localisations of spectra are exactly the finite ones that are the identity away from finitely many primes.
Ohkawa’s theorem: there is a set of Bous- field classes
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A strong finiteness condition for smashing localisations
The localisation L^f_n is compactly central, and the compactly central localisations of spectra are exactly the finite ones that are the identity away from finitely many primes.