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  "body": "\n## Definition\n\nFix a [[prime number|01826KMG2H]] $p$ and work with $p$-local [[spectra|1GHF4EKBXS]] in the [[stable homotopy category|26GTNMAF7B]].\nFor a spectrum $E$, a spectrum $X$ is *$E$-acyclic* if $E \\wedge X \\simeq 0$, and *$E$-local* if every map from an $E$-acyclic spectrum to $X$ is null.\nThe *Bousfield localization* $X \\to L_E X$ is the map to an $E$-local spectrum whose fibre is $E$-acyclic; it exists and is unique up to equivalence.\nFor $n \\ge 1$, let $E(n)$ be the Johnson–Wilson spectrum, with $\\pi_* E(n) = \\Z_{(p)}[v_1, \\ldots, v_n, v_n^{-1}]$, a quotient of [[Brown–Peterson cohomology|S8S72AA9C8]] with $v_n$ inverted; let $E(0) = H\\Q$.\nWrite $L_n = L_{E(n)}$.\nSince $E(n-1)$-local spectra are $E(n)$-local, there are natural maps $L_n X \\to L_{n-1} X$.\nThe *chromatic tower* of $X$ is\n$$X \\to \\cdots \\to L_n X \\to L_{n-1} X \\to \\cdots \\to L_0 X = X \\wedge H\\Q.$$\nLet $K(n)$ be the $n$-th Morava K-theory, with $\\pi_* K(n) = \\mathbb{F}_p[v_n, v_n^{-1}]$ and $\\abs{v_n} = 2(p^n - 1)$.\n*Chromatic homotopy theory* is the study of spectra through their chromatic towers, their *monochromatic layers* $M_n X = \\operatorname{fib}(L_n X \\to L_{n-1} X)$, and their localizations $L_{K(n)} X$.\n\n## Remarks\n\nExample: for the $p$-local sphere spectrum $S$, the layer $L_0 S = H\\Q$ records the rational stable homotopy, which is $\\Q$ in degree $0$ only.\nThe layer $L_1 S$ is computed from K-theory and contains the image of the J-homomorphism.\n\nNon-example: the Eilenberg–MacLane spectrum $H\\mathbb{F}_p$ is not $E(n)$-local for any $n$.\nIndeed $E(n) \\wedge H\\mathbb{F}_p \\simeq 0$, so $H\\mathbb{F}_p$ is a nonzero $E(n)$-acyclic spectrum, and its identity map is not null.\nSo all terms of its chromatic tower vanish.\n\nBy the chromatic convergence theorem of Hopkins and Ravenel, a finite $p$-local spectrum is the homotopy limit of its chromatic tower.\nThe example of $H\\mathbb{F}_p$ shows that this fails without finiteness.\n",
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