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  "body": "\n## Counterexample\n\nFix a prime $p$ and $n \\ge 1$.\nLet $X$ be a finite $p$-local [[spectrum|1GHF4EKBXS]] of type $n$.\nLet $v \\colon \\Sigma^d X \\to X$ be a $v_n$ self-map, as given by the periodicity theorem among the [[Ravenel conjectures|B8XP2K7RS1]].\nThe telescope of $v$ is the homotopy colimit\n$$v^{-1} X = \\operatorname{hocolim}\\left( X \\xrightarrow{v} \\Sigma^{-d} X \\xrightarrow{v} \\Sigma^{-2d} X \\to \\cdots \\right).$$\nThe telescope conjecture asserted that the natural map $v^{-1} X \\to L_{K(n)} X$ is an equivalence, where $L_{K(n)}$ is Bousfield localization at the $n$-th Morava K-theory.\nThe conjecture is false for every prime $p$ and every height $n \\ge 2$.\nIn 2023 Burklund, Hahn, Levy and Schlank showed that for such $p$ and $n$ the map $v^{-1} X \\to L_{K(n)} X$ is not an equivalence.\n\n## Remarks\n\nThe telescope $v^{-1} X$ does not depend on the choice of $v$, and its Bousfield class depends only on $n$; the localization at it is written $L_{T(n)}$.\nSo the conjecture says $L_{T(n)} = L_{K(n)}$.\nAt height $1$ the conjecture is true: Mahowald proved it for $p = 2$ and Miller for $p$ odd.\nThe counterexample of Burklund, Hahn, Levy and Schlank uses algebraic K-theory to build $T(n)$-local spectra that are not $K(n)$-local.\nIt shows that the homotopy of telescopes is genuinely larger than what $K(n)$-local computations see.\n\n## Explanations\n\n### L1 human\n\nA finite spectrum of height $n$ has a periodic self-map, and inverting it gives a \"telescope\".\nRavenel conjectured that the telescope is the same as the localization at Morava K-theory, which is much easier to compute.\nThis is true at heights $0$ and $1$.\nAt every height from $2$ on it is false, so the telescope holds information that Morava K-theory does not see.\n",
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    "name": "Telescope conjecture",
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  "msc": [
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  "name": "Telescope conjecture",
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