Higher Semiadditive Character Theory

Abstract

We introduce and develop the theory of semiadditive characters in the higher semiadditive setting, generalizing both the $T(n)$-local monoidal character and the $K(t)$-local transchromatic character. These are natural transformations compatible with restriction and transfer maps along $π$-finite spaces, with an $(n-t)$-fold $p$-typical free loop space correction built into the target. We show that every $\infty$-commutative monoid admits a universal $(n-t)$-fold character. This universal character has several strong structural properties: it exhibits blue shift, satisfies higher cyclotomic descent, and is compatible with the semiadditive Fourier transform. We compute it for an arbitrary $K(n)$-local object and show that, for Morava $E$-theory, it recovers the $K(t)$-local transchromatic character. By functoriality, the universal character carries a natural action of the profinite group $\mathrm{GL}_{n-t}(\mathbb{Z}_p)$. When $t=0$, the fixed points of this action recover rationalization. As a consequence, we derive an explicit description of $L_{\mathbb{Q}}(S^A_{K(n)})$ for every $π$-finite space $A$ and compute the ring of rational $K(n)$-local power operations.

Disclosure

“hout. I am also grateful to my academic brother, Shai Keidar, for numerous insightful discussions and for reading earlier drafts of this paper. I thank Beckham Myers, Shay Ben-Moshe, and Lior Yanovski for their comments on previous drafts. LLMs were used to improve the writing. All mathematical content, and any errors, are my own. 2 Higher Semiadditive Characters This section defines and records the basic properties of F -characters that we will use throughout the paper. In”

PDF page 12

Classification

Rewriting existing author-written text

Multiplier

4

Verified

2026-08-24T00:34:15.228Z

Structural counts