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Swan induction for the finite-local sphere

Primarily AI-generated textHuman understanding: some partsmath.AT — Algebraic Topologymath.KT — K-Theory and Homology

Contributed by Akhil Mathew ↗

SubmitterAkhil Mathew

Version 1 / Sep 27, 2026 / CC BY 4.0

Abstract

We establish rational Swan induction for ordinary perfect modules over the finite-local sphere Lnp,fSL_n^{p,f}\mathbb S. For a finite abelian ambient group, subgroups of pp-rank at most n+1n+1 suffice, and this bound is sharp. The proof combines cyclic homotopy fixed points in telescopic spectra with the isotropy filtration of a chromatic quotient of finite genuine spectra. The result proves Conjecture~7.22 of Clausen--Mathew--Naumann--Noel for Morava EE-theory and gives a new proof of their chromatic upper bound for the algebraic KK-theory of Lnp,fSL_n^{p,f}\mathbb S-linear categories. The quotient argument also produces finite complexes realizing the induction relations. We formulate the problem of explicit realizations and give two geometric models at the prime two.

Provenance statement

I believe the arguments and strategy to be correct; however, I have not yet fully checked and understood all the details. The paper was generated with ChatGPT, and I also asked Claude to proofread/check it.

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Version history

  1. v1Initial depositCurrentSep 27, 2026