A note on higher topological Hochschild homology

TL;DR

This paper investigates how higher topological Hochschild homology can detect increased chromatic heights in ring spectra, revealing deeper chromatic redshift phenomena through homotopy fixed points.

Contribution

It introduces a method to detect higher chromatic redshift using homotopy fixed points of higher topological Hochschild homology, extending previous understanding.

Findings

Homotopy fixed points of higher topological Hochschild homology detect v_{n+k}-elements.

The approach extends chromatic redshift detection beyond negative topological cyclic homology.

Provides new tools for understanding chromatic phenomena in algebraic K-theory.

Abstract

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.