Examples of chromatic redshift 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 $n\ge 1$ 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 the implicit prime $p$, has nontrivial $T(n)$-localization.