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

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 $\mathrm{BP}⟨n⟩$, we prove that the $T(n+1)$-localized algebraic $K$-theory of $\mathrm{BP}⟨n{⟩}^{h\mathbb{Z}}$ 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 $T(2{)}_{\ast }\mathrm{K}(R)$, for $R$ certain finite Galois extensions of the $K(1)$-local sphere. We show for $p\ge 5$ that the algebraic $K$-theory of the $K(1)$-local sphere is asymptotically $L_2^f$-local.

100 pages. Comments very welcome