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arxiv: 2310.17459 · v1 · pith:36L5BN5Rnew · submitted 2023-10-26 · 🧮 math.AT · math.KT

K-theoretic counterexamples to Ravenel's telescope conjecture

Robert Burklund , Jeremy Hahn , Ishan Levy , Tomer M. Schlank This is my paper
classification 🧮 math.AT math.KT
keywords localmathbbalgebraicmathrmtheorygaloislanglelocalized
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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.

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