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arXiv · 2309.07123

Descent and cyclotomic redshift for chromatically localized algebraic K-theory

Shay Ben-Moshe·Shachar Carmeli·Tomer M. Schlank·Lior Yanovski
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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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Shay Ben-Moshe, Shachar Carmeli, Tomer M. Schlank, Lior Yanovski. 2023-09-13. Descent and cyclotomic redshift for chromatically localized algebraic K-theory. https://doi.org/10.1090/jams%2F1052

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