{"id":"hexagon/2609.00112","pdf":"https://hexagonmath.org/pdf/2609.00112v1","url":"https://hexagonmath.org/abs/2609.00112v1","tags":["math.KT","msc:55P42","msc:19A22","msc:55P91","msc:55P60","msc:19D10","Burnside rings","Swan induction","chromatic homotopy theory"],"score":0,"title":"Swan induction for the finite-local sphere","domain":"math.AT","origin":"ai-generated","source":"hexagon","authors":["Akhil Mathew"],"reviews":0,"subject":"math.AT","summary":"We establish rational Swan induction for ordinary perfect modules over the finite-local sphere $L_n^{p,f}\\mathbb S$. For a finite abelian ambient group, subgroups of $p$-rank at most $n+1$ 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 $E$-theory and gives a new proof of their chromatic upper bound for the algebraic $K$-theory of $L_n^{p,f}\\mathbb S$-linear categories. The…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-27","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}