Abstract
Abstract Using higher descent for chromatically localized algebraic $K$-theory, we show that the higher semiadditive cardinality of a $\pi $-finite $p$-space $A$ at the Lubin–Tate spectrum $E_{n}$ is equal to the higher semiadditive cardinality of the free loop space $LA$ at $E_{n-1}$. By induction, it is thus equal to the homotopy cardinality of the $n$-fold free loop space $L^{n} A$. We explain how this allows one to bypass the Ravenel–Wilson computation in the proof of the $\infty $-semi-additivity of the ${T(n)} $-local categories. “Cardinalis sinuatus – the Pyrrhuloxia” by Dick Culbert licensed under CC BY 2.0.
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