Adjunction of roots, algebraic \(K\)-theory and chromatic redshift

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-T...

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Bibliographic Details
Published in:arXiv.org 2023-10
Main Authors: Ausoni, Christian, Haldun Özgür Bayındır, Moulinos, Tasos
Format: Article
Language:English
Subjects:
Algebra
Red shift
Online Access:Get full text
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Summary: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 \(V(1)_*K(ko_p)\) 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.
ISSN:2331-8422