Chromatic homotopy theory

Definition

Fix a prime number $p$ and work with $p$-local spectra in the stable homotopy category. For 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. The 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. For $n \ge 1$, 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 with $v_n$ inverted; let $E(0) = H\Q$. Write $L_n = L_{E(n)}$. Since $E(n-1)$-local spectra are $E(n)$-local, there are natural maps $L_n X \to L_{n-1} X$. The chromatic tower of $X$ is $$X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X = X \wedge H\Q.$$ Let $K(n)$ be the $n$-th Morava K-theory, with $\pi_* K(n) = \mathbb{F}_p[v_n, v_n^{-1}]$ and $\abs{v_n} = 2(p^n - 1)$. 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 $L_{K(n)} X$.