Descent and cyclotomic redshift for chromatically localized algebraic 𝐾-theory

Ben-Moshe, Shay; Carmeli, Shachar; Schlank, Tomer; Yanovski, Lior

doi:10.1090/jams/1052

Descent and cyclotomic redshift for chromatically localized algebraic $K$-theory
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by Shay Ben-Moshe, Shachar Carmeli, Tomer M. Schlank and Lior Yanovski;

J. Amer. Math. Soc.

DOI: https://doi.org/10.1090/jams/1052

Published electronically: November 18, 2024

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Abstract:

We prove that $T(n+1)$-localized algebraic $K$-theory satisfies descent for $\pi$-finite $p$-group actions on stable $\infty$-categories of chromatic height up to $n$, extending a result of Clausen–Mathew–Naumann–Noel for finite $p$-groups. Using this, we show that it sends $T(n)$-local Galois extensions to $T(n+1)$-local Galois extensions. Furthermore, we show that it sends cyclotomic extensions of height $n$ to cyclotomic extensions of height $n+1$, extending a result of Bhatt–Clausen–Mathew for $n=0$. As a consequence, we deduce that $K(n+1)$-localized $K$-theory satisfies hyperdescent along the cyclotomic tower of any $T(n)$-local ring. Counterexamples to such cyclotomic hyperdescent for $T(n+1)$-localized $K$-theory were constructed by Burklund, Hahn, Levy and the third author, thereby disproving the telescope conjecture.

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