Examples of chromatic redshift in algebraic $K$-theory

TL;DR

This paper presents a straightforward method to detect chromatic redshift in algebraic $K$-theory of $ ext{E}_ty$-ring spectra, with applications to Lubin-Tate theories and iterated algebraic $K$-theory of fields.

Contribution

It introduces a simple argument for detecting chromatic redshift and applies it to show nontrivial localizations in specific algebraic $K$-theory cases.

Findings

$K(E_n)$ has nontrivial $T(n+1)$-localization for $n q 1$

$K^{(n)}(k)$ has nontrivial $T(n)$-localization for fields of characteristic not $p$

Provides a new tool for understanding chromatic phenomena in algebraic $K$-theory

Abstract

We give a simple argument to detect chromatic redshift in the algebraic $K$-theory of $\mathbb{E}_{\infty}$-ring spectra and give two applications: we show for $n\geq 1$ that $K(E_n)$, the algebraic $K$-theory of any height $n$ Lubin-Tate theory, has nontrivial $T(n+1)$-localization, and that $K^{(n)}(k)$, the $n$-fold iterated algebraic $K$-theory of a field $k$ of characteristic different from $p$, has nontrivial $T(n)$-localization.