The Chromatic Fourier Transform

TL;DR

This paper develops a comprehensive theory of higher semiadditive Fourier transforms, unifying classical and modern dualities in algebraic topology, and extends these concepts to new settings including spectra and Galois extensions.

Contribution

It generalizes duality theories in algebraic topology to broader contexts, including spectra and Galois extensions, and introduces a categorified equivalence of symmetric monoidal categories.

Findings

Unified classical and modern Fourier dualities

Extended duality to all finite spectra and computed discrepancy spectrum

Established new results on Galois extensions and categorified dualities

Abstract

We develop a general theory of higher semiadditive Fourier transforms that includes both the classical discrete Fourier transform for finite abelian groups at height $n=0$, as well as a certain duality for the $E_n$-(co)homology of $\pi$-finite spectra, established by Hopkins and Lurie, at heights $n\ge 1$. We use this theory to generalize said duality in three different directions. First, we extend it from $\mathbb{Z}$-module spectra to all (suitably finite) spectra and use it to compute the discrepancy spectrum of $E_n$. Second, we lift it to the telescopic setting by replacing $E_n$ with $T(n)$-local higher cyclotomic extensions, from which we deduce various results on affineness, Eilenberg--Moore formulas and Galois extensions in the telescopic setting. Third, we categorify their result into an equivalence of two symmetric monoidal $\infty$-categories of local systems of $K(n)$-local $E_n$-modules, and relate it to (semiadditive) redshift phenomena.