math.AT — Algebraic Topology

Swan induction for the finite-local sphere

Contributed by Akhil Mathew

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 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.