chromatic localization

The Bousfield localization of spectra $L_{E(n)}$ at $n$th Morava E-theory is called *chromatic localization*. The tower of chromatic localizations as $n$ ranges is called the *chromatic tower*, leading to the *chromatic filtration*. This is the subject of chromatic homotopy theory.

Chromatic localization on complex oriented cohomology theories is like the restriction to the closed substack

$\mathcal{M}_{FG}^{\leq n+1}
\hookrightarrow
\mathcal{M}_{FG} \times Spec \mathbb{Z}_{(p)}$

of the moduli stack of formal groups on those of height $\geq n+1$.

(e.g. Lurie, lect 22, above theorem 1)

In this way the localization tower at the Morava E-theories exhibits the chromatic filtration in chromatic homotopy theory.

chromatic level | complex oriented cohomology theory | E-∞ ring/A-∞ ring | real oriented cohomology theory |
---|---|---|---|

0 | ordinary cohomology | Eilenberg-MacLane spectrum $H \mathbb{Z}$ | HZR-theory |

0th Morava K-theory | $K(0)$ | ||

1 | complex K-theory | complex K-theory spectrum $KU$ | KR-theory |

first Morava K-theory | $K(1)$ | ||

first Morava E-theory | $E(1)$ | ||

2 | elliptic cohomology | elliptic spectrum $Ell_E$ | |

second Morava K-theory | $K(2)$ | ||

second Morava E-theory | $E(2)$ | ||

algebraic K-theory of KU | $K(KU)$ | ||

3 …10 | K3 cohomology | K3 spectrum | |

$n$ | $n$th Morava K-theory | $K(n)$ | |

$n$th Morava E-theory | $E(n)$ | BPR-theory | |

$n+1$ | algebraic K-theory applied to chrom. level $n$ | $K(E_n)$ (red-shift conjecture) | |

$\infty$ | complex cobordism cohomology | MU | MR-theory |

- Jacob Lurie,
*Chromatic Homotopy Theory*, Lecture series 2010 (lecture notes)

- Mark Hovey, Hal Sadofsky,
*Invertible spectra in the $E(n)$-local stable homotopy category*(pdf)