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Finite Galois Descent in Telescopic Algebraic K-Theory

Author: Wei Yang · Added: September 24, 2026

Article date: September 2026.

Abstract

Fix a prime $p$ and an integer $n\geq0$ . We prove that every finite $G$ -Galois extension $R\to S$ of $T(n)$ -local commutative ring spectra satisfies

L_{T(n+1)}K(R)\simeq \bigl(L_{T(n+1)}K(S)\bigr)^{hG}.

For $n\geq1$ , we establish prime-to- $p$ cyclic descent by a comparison of crossed-product Hochschild sectors, realized by a morphism of cyclotomic spectra. For $n=0$ , we prove that even invertible modules of order prime to $p$ have class $1$ in the zeroth homotopy group of height-one localized $K$ -theory of arbitrary rational rings. This follows from invariance under affine torsors for even invertible modules and a finite cell argument. The family-completeness theorem of Clausen-Mathew-Naumann-Noel then gives descent for every finite group in both cases.