Rixin Fang | Searcharxiv

arXiv2026

A note on higher topological Hochschild homology

Chromatic redshift phenomena suggest that algebraic K-theory increases the height of a commutative ring spectrum by one. In many cases, the chromatic redshift is already detected by negative topological cyclic homology. This paper explores higher chromatic redshift via homotopy fixed points spectrum of higher topological Hochschild homology. Specifically, starting from a commutative ring spectrum that detects $v_n$-elements, the homotopy fixed points spectrum of higher topological Hochschild homology of it detects $v_{n+k}$-elements, with $k$ greater than one.

math.AT↗

arXiv2025

Algebraic K-theory of finite algebras over higher local fields

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.

math.KT↗