[Paper Review] Descent and cyclotomic redshift for chromatically localized algebraic K-theory

Shay Ben-Moshe, Shachar Carmeli+2 more|arXiv (Cornell University)|Sep 13, 2023

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes that $T(n+1)$-localized algebraic $K$-theory satisfies higher descent for $\pi$-finite $p$-group actions on $L_n^f$-local stable $\infty$-categories, extends cyclotomic redshift to cyclotomic extensions of height $n$, and proves that $K(n+1)$-localized $K$-theory satisfies hyperdescent along the cyclotomic tower of any $T(n)$-local ring. These results confirm a refined form of the redshift conjecture and resolve a key obstruction to the telescope conjecture via counterexamples to cyclotomic hyperdescent.

ABSTRACT

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.

Motivation & Objective

• To extend the descent properties of $T(n+1)$-localized algebraic $K$-theory to $\pi$-finite $p$-group actions on $L_n^f$-local categories.
• To establish that $T(n+1)$-localized $K$-theory sends $T(n)$-local Galois extensions to $T(n+1)$-local Galois extensions.
• To prove that $T(n+1)$-localized $K$-theory sends cyclotomic extensions of height $n$ to cyclotomic extensions of height $n+1$, extending Bhatt–Clausen–Mathew for $n=0$.
• To show that $K(n+1)$-localized $K$-theory satisfies hyperdescent along the cyclotomic tower of any $T(n)$-local ring.
• To provide counterexamples to cyclotomic hyperdescent for $T(n+1)$-localized $K$-theory, thereby disproving the telescope conjecture.

Proposed method

• Extends Clausen–Mathew–Naumann–Noel's descent result from finite $p$-groups to $\pi$-finite $p$-groups using higher categorical descent techniques.
• Applies the theory of monochromatic categories and categorical localization to analyze $L_n^f$-local stable $\infty$-categories.
• Uses the Fourier transform and Kummer theory in the context of cyclotomic extensions to relate $K$-theory of $R[\omega_{p^{(-)}}^{(n)}]$ to cyclotomic sheaves.
• Applies the $d^*$ and $d_*$ functors on the site of $\mathbb{Z}_p$-sets to relate $K$-theory to hypersheafification and cyclotomic completion.
• Employs the notion of cyclotomic completion and $\mathbb{S}_{T(n+1)}[\omega_{p^\infty}^{(n+1)}]$-localization to characterize hypersheaves.
• Uses the equivalence between cyclotomic completeness and $\mathbb{S}_{T(n+1)}[\overline{\omega}_{p^\infty}^{(n+1)}]$-localization to establish hyperdescent.

Experimental results

Research questions

• RQ1 Does $T(n+1)$-localized algebraic $K$-theory satisfy descent for $\pi$-finite $p$-group actions on $L_n^f$-local categories?
• RQ2 Does $T(n+1)$-localized $K$-theory send $T(n)$-local Galois extensions to $T(n+1)$-local Galois extensions?
• RQ3 Does $T(n+1)$-localized $K$-theory send cyclotomic extensions of height $n$ to cyclotomic extensions of height $n+1$?
• RQ4 Does $K(n+1)$-localized $K$-theory satisfy hyperdescent along the cyclotomic tower of a $T(n)$-local ring?
• RQ5 Can counterexamples to cyclotomic hyperdescent for $T(n+1)$-localized $K$-theory be constructed, and what do they imply for the telescope conjecture?

Key findings

• The paper proves that $T(n+1)$-localized algebraic $K$-theory satisfies descent for $\pi$-finite $p$-group actions on $L_n^f$-local stable $\infty$-categories, generalizing prior results for finite $p$-groups.
• It establishes that $T(n+1)$-localized $K$-theory sends $T(n)$-local Galois extensions to $T(n+1)$-local Galois extensions, confirming a key prediction of the redshift philosophy.
• It shows that $T(n+1)$-localized $K$-theory sends cyclotomic extensions of height $n$ to cyclotomic extensions of height $n+1$, extending Bhatt–Clausen–Mathew’s result for $n=0$.
• It proves that $K(n+1)$-localized $K$-theory satisfies hyperdescent along the cyclotomic tower of any $T(n)$-local ring, via hypersheafification and cyclotomic completion.
• It constructs counterexamples to cyclotomic hyperdescent for $T(n+1)$-localized $K$-theory, thereby disproving the telescope conjecture.
• The results imply that $K_{T(n+1)}(R[\overline{\omega}_{p^{(-)}}^{(n)}])$ is both a hypersheaf and level-wise cyclotomically complete, with the target of the $d^*$-map being the hypersheafification and cyclotomic completion of the source.