Adjunction of roots, algebraic $K$-theory and chromatic redshift

TL;DR

The paper introduces a method for adjoining roots to $E_1$-rings, analyzes the impact on algebraic $K$-theory and chromatic redshift, and applies this to Lubin-Tate spectra, providing new insights and proofs.

Contribution

It defines root adjunction for $E_1$-rings, explores its effects on THH and $K$-theory, and demonstrates redshift preservation, including a new proof for Lubin-Tate spectra.

Findings

Root adjunction is log-THH-étale under certain conditions.

The induced $K$-theory map is a wedge summand inclusion.

Chromatic redshift property is preserved under root adjunction.

Abstract

Given an $E_1$-ring $A$ and a class $a \in \pi_{mk}(A)$ satisfying a suitable hypothesis, we define a map of $E_1$-rings $A\to A(\sqrt[m]{a})$ realizing the adjunction of an $m$th root of $a$. We define a form of logarithmic THH for $E_1$-rings, and show that root adjunction is log-THH-\'etale for suitably tamely ramified extension, which provides a formula for THH$(A(\sqrt[m]{a}))$ in terms of THH and log-THH of $A$. If $A$ is connective, we prove that the induced map $K(A) \to K(A(\sqrt[m]{a}))$ in algebraic $K$-theory is the inclusion of a wedge summand. Using this, we obtain $V(1)_*K(ko_p)$ for $p>3$ and also, we deduce that if $K(A)$ exhibits chromatic redshift, so does $K(A(\sqrt[m]{a}))$. We interpret several extensions of ring spectra as examples of root adjunction, and use this to obtain a new proof of the fact that Lubin-Tate spectra satisfy the redshift conjecture.