In algebraic topology, the chromatic convergence theorem states the homotopy limit of the chromatic tower (defined below) of a finite p-local spectrum ${\displaystyle X}$  is ${\displaystyle X}$  itself. The theorem was proved by Hopkins and Ravenel.

Statement edit

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 edit

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.

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

• 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.

• 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.