Algebraic K-theory of finite algebras over higher local fields
Rixin Fang
TL;DR
This work analyzes the algebraic K-theory of truncated polynomial algebras over higher-height local field analogs in chromatic homotopy theory, focusing on Segal-type conjectures and Lichtenbaum–Quillen properties. By constructing finite $ ext{BP} angle n angle$-algebras and applying THH/TC descent, Frobenius computations, and spectral sequences, it shows Segal fails for $ ext{BP} angle n angle[x]/x^e$ when $e>1$ while remaining valid in several MU and BP variants; it also demonstrates that the Lichtenbaum–Quillen property fails at higher height, yet weak redshift persists. The paper develops a comprehensive framework of $ ext{E}_3$-algebras, descent theories, and height-dependent phenomena to illuminate how chromatic localization interacts with singularities in algebraic K-theory. These results sharpen our understanding of the interplay between Segal conjecture, redshift, and K-theory in the presence of truncated polynomial extensions, and they point to further avenues via cyclic decompositions and syntomic-type descent. Overall, the findings map a nuanced landscape where certain higher-height constructions violate LQ-type boundedness and Segal-type expectations, while other, simpler variants retain these properties, highlighting delicate chromatic behavior.
Abstract
It is known that the truncated Brown--Peterson spectra can be equipped with a certain nice algebra structure, by the work of J. Hahn and D. Wilson, and these ring spectra can be viewed as rings of integers of local fields in chromatic homotopy theory. Furthermore, they satisfy both Rognes' redshift conjecture and the Lichtenbaum--Quillen property. For lower-height cases, the K-theory of the truncated polynomial algebras over these ring spectra is well understood through the work of L. Hesselholt, I. Madsen, and others. In this paper, we demonstrate that the Segal conjecture fails for truncated polynomial algebras over higher chromatic local fields, and consequently, the Lichtenbaum--Quillen property fails. However, the weak redshift conjecture remains valid. Additionally, we provide some other examples where Segal conjecture holds.