Chromatic homotopy theory

In mathematics, chromatic homotopy theory is a subfield of

${\displaystyle X}$ is ${\displaystyle X}$ itself. The theorem was proved by Hopkins and Ravenel.

Statement

Let ${\displaystyle L_{E(n)}}$ denotes the Bousfield localization with respect to the Morava E-theory and let ${\displaystyle X}$ be a finite, ${\displaystyle p}$-local spectrum. Then there is a tower associated to the localizations

${\displaystyle \cdots \rightarrow L_{E(2)}X\rightarrow L_{E(1)}X\rightarrow L_{E(0)}X}$

called the chromatic tower, such that its homotopy limit is homotopic to the original spectrum ${\displaystyle X}$.

The stages in the tower above are often simplifications of the original spectrum. For example, ${\displaystyle L_{E(0)}X}$ is the rational localization and ${\displaystyle L_{E(1)}X}$ is the localization with respect to p-local K-theory.

Stable homotopy groups

In particular, if the ${\displaystyle p}$-local spectrum ${\displaystyle X}$ is the stable ${\displaystyle p}$-local sphere spectrum ${\displaystyle \mathbb {S} _{(p)}}$, then the homotopy limit of this sequence is the original ${\displaystyle p}$-local sphere spectrum. This is a key observation for studying stable homotopy groups of spheres using chromatic homotopy theory.

See also

• Elliptic cohomology
• Redshift conjecture
• Ravenel conjectures
• Moduli stack of formal group laws
• Chromatic spectral sequence
• Adams-Novikov spectral sequence

References

• Lurie, J. (2010). "Chromatic Homotopy Theory". 252x (35 lectures). Harvard University.
• Lurie, J. (2017–2018). "Unstable Chomatic Homotopy Theory". 19 Lectures. Institute for Advanced Study.

External links

• http://ncatlab.org/nlab/show/chromatic+homotopy+theory
• Hopkins, M. (1999). "Complex Oriented Cohomology Theory and the Language of Stacks" (PDF). Archived from the original (PDF) on 2020-06-20.
• "References, Schedule and Notes". Talbot 2013: Chromatic Homotopy Theory. MIT Talbot Workshop. 2013.

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