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\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 the implicit prime p , has nontrivial T(n) -localization.