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

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-étale 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 $V(1{)}_{\ast }K(k{o}_p)$ 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.

Substantial revision