Chromatic Cardinalities via Redshift

TL;DR

This paper establishes a recursive relationship between higher semiadditive cardinalities of certain spaces at different Lubin-Tate spectra, enabling a new approach to proving $ ext{infty}$-semiadditivity in chromatic homotopy theory.

Contribution

It introduces a novel method using higher descent to relate cardinalities across chromatic levels, bypassing complex classical computations.

Findings

Higher semiadditive cardinality of a $ ext{pi}$-finite $p$-space at $E_n$ equals that of its free loop space at $E_{n-1}$.

This recursive relation simplifies proofs of $ ext{infty}$-semiadditivity in $T(n)$-local categories.

The approach leverages higher descent to connect homotopy cardinalities across chromatic heights.

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 $\infty$-semiadditivity of the $T(n)$-local categories.