Adjunction of roots, algebraic K-theory and chromatic redshift

2022/11/30 by Christian Ausoni, Haldun Özgür Bayındır, Ausoni, Christian +3 · 2 citations
Mathematics · #19D99 #55P99 #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT)

paper · pdf · doi:10.48550/arxiv.2211.16929

openalex publication_date 2022/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given an E₁-ring A and a class a ∈ π_mk(A) satisfying a suitable hypothesis, we define a map of E₁-rings A→ A(√[m]a) realizing the adjunction of an mth root of a. We define a form of logarithmic THH for E₁-rings, and show that root adjunction is log-THH-étale for suitably tamely ramified extension, which provides a formula for THH(A(√[m]a)) in terms of THH and log-THH of A. If A is connective, we prove that the induced map K(A) → K(A(√[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(√[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.

Citations

• Algebraic K-theory of topological K-theory
• Bökstedt periodicity and quotients of DVRs
• On topological cyclic homology

Cited by

• Towards the classification of DGAs with polynomial homology
• On the Residue Sequence in Logarithmic Topological Cyclic Homology

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