= Chromatic homotopy theory
{title2=$K(n)_*(X)$}
Chromatic homotopy theory organizes <stable homotopy theory> by a <prime number> $p$ and a hierarchy of heights $n$. Its main probes include <Morava K-theory>, whose standard coefficient ring for $n\ge1$ is
$$
K(n)_*\cong\mathbb F_p[v_n,v_n^{-1}],\qquad |v_n|=2(p^n-1).
$$
The index is the height of the associated <formal group law>. <Rational homology> plays the height-zero role. These different probes detect different parts of a <spectrum (topology)>, rather than replacing all stable homotopy information by one ordinary <homology group>.<Morava E-Theory and Morava K-Theory by Jacob Lurie>[Lecture 22, Definition 5 and coefficient calculation, page 3]{ref}
A basic chromatic question asks which heights detect a given finite <spectrum (topology)>: its type is the least $n$ with $K(n)_*(X)\ne0$, when such an $n$ exists. Rational information already detects the <sphere spectrum> at height zero, whereas purely torsion phenomena require other probes. The chromatic viewpoint makes periodic patterns and <localization of spectra> into organizing tools for difficult stable computations.
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