Thom spectrum

Fix a prime \(p\). The chromatic tower of a Thom spectrum \(X\) is the tower of Bousfield localization \begin{align*} X \rightarrow \cdots \rightarrow X _ { E ( n ) } \rightarrow X _ { E ( n - 1 ) } \rightarrow \cdots \rightarrow X _ { E ( 0 ) } \end{align*} where \(E(n)\) is the \(n\)th Johnson-Wilson spectrum \((E(0) = { \mathsf{H} }{\mathbb{Q}}\), by convention) with \begin{align*} E ( n ) _ { * } = \mathbb { Z } _ { ( p ) } \left[ v _ { 1 } , \dots , v _ { n - 1 } , v _ { n } ^ { \pm } \right] \end{align*} The fibers of the chromatic tower \begin{align*} M _ { n } X \rightarrow X _ { E ( n ) } \rightarrow X _ { E ( n - 1 ) } \end{align*} are called the monochromatic layers.