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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 $n\ge 1$ isThe 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, rather than replacing all stable homotopy information by one ordinary homology group.^[ref]

A basic chromatic question asks which heights detect a given finite spectrum: its type is the least $n$ with $K(n{)}_{\ast }(X)\ne 0$, 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.

Given a topological spectrum $E$, its localization of spectra sends $X$ to an $E$-local topological spectrum $L_{E}X$ through a map $X\to L_{E}X$ inducing an isomorphism on $E$-homology. Here $E$-local means that every map from an $E$-acyclic topological spectrum to $L_{E}X$ is nullhomotopic; $A$ is $E$-acyclic when $E\wedge A$ is trivial. This is Bousfield localization.^[ref]

The map is universal for maps from $X$ to $E$-local objects. For rational homology, localization rationalizes homotopy groups: ${\pi }_j(L_{H\mathbb{Q}}X)={\pi }_j(X)\otimes \mathbb{Q}$. Thus the sphere spectrum loses its finite torsion information and retains $\mathbb{Q}$ in degree zero. Localization deliberately preserves the information seen by a chosen generalized homology theory while discarding information invisible to it.