Chromatic Cardinalities via Redshift

Using higher descent for chromatically localized algebraic K-theory, we show that the higher semiadditive cardinality of a π-finite p-space A at the Lubin–Tate spectrum En is equal to the higher semiadditive cardinality of the free loop space LA at En−1. By induction, it is thus equal to the homotopy cardinality of the n-fold free loop space LⁿA. We explain how this allows one to bypass the Ravenel–Wilson computation in the proof of the ∞-semi-additivity of the T(n)-local categories.