Descent and cyclotomic redshift for chromatically localized algebraic K-theory
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| Format: | Preprint |
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2023
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| _version_ | 1866929604397105152 |
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| author | Ben-Moshe, Shay Carmeli, Shachar Schlank, Tomer M. Yanovski, Lior |
| author_facet | Ben-Moshe, Shay Carmeli, Shachar Schlank, Tomer M. Yanovski, Lior |
| contents | We prove that $T(n+1)$-localized algebraic $K$-theory satisfies descent for $π$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_07123 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Descent and cyclotomic redshift for chromatically localized algebraic K-theory Ben-Moshe, Shay Carmeli, Shachar Schlank, Tomer M. Yanovski, Lior K-Theory and Homology Algebraic Topology 19D99, 55P42 We prove that $T(n+1)$-localized algebraic $K$-theory satisfies descent for $π$-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. |
| title | Descent and cyclotomic redshift for chromatically localized algebraic K-theory |
| topic | K-Theory and Homology Algebraic Topology 19D99, 55P42 |
| url | https://arxiv.org/abs/2309.07123 |