$K$-theoretic counterexamples to Ravenel's telescope conjecture

TL;DR

This paper demonstrates that at each prime p and height n+1 ≥ 2, the localized algebraic K-theory of certain spectra diverges from expected localizations, revealing failures of several descent properties and providing explicit computations for specific cases.

Contribution

It provides the first examples of K-theoretic counterexamples to Ravenel's telescope conjecture at all primes and heights, and analyzes the failure of descent properties in localized algebraic K-theory.

Findings

Telescopic and chromatic localizations differ at each prime p and height n+1 ≥ 2.

Galois hyperdescent, A^1-invariance, and nil-invariance fail for localized algebraic K-theory.

Complete computations of T(2)_*K(R) for certain Galois extensions of the K(1)-local sphere at p ≥ 7.

Abstract

At each prime $p$ and height $n+1 \ge 2$, we prove that the telescopic and chromatic localizations of spectra differ. Specifically, for $\mathbb{Z}$ acting by Adams operations on $\mathrm{BP}\langle n \rangle$, we prove that the $T(n+1)$-localized algebraic $K$-theory of $\mathrm{BP}\langle n \rangle^{h\mathbb{Z}}$ is not $K(n+1)$-local. We also show that Galois hyperdescent, $\mathbb{A}^1$-invariance, and nil-invariance fail for the $K(n+1)$-localized algebraic $K$-theory of $K(n)$-local $\mathbb{E}_{\infty}$-rings. In the case $n=1$ and $p \ge 7$ we make complete computations of $T(2)_*\mathrm{K}(R)$, for $R$ certain finite Galois extensions of the $K(1)$-local sphere. We show for $p\geq 5$ that the algebraic $K$-theory of the $K(1)$-local sphere is asymptotically $L_2^{f}$-local.