• Source: Chromatic homotopy theory

In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups. In this picture, theories are classified in terms of their "chromatic levels"; i.e., the heights of the formal groups that define the theories via the Landweber exact functor theorem. Typical theories it studies include: complex K-theory, elliptic cohomology, Morava K-theory and tmf.




Chromatic convergence theorem


In algebraic topology, the chromatic convergence theorem states the homotopy limit of the chromatic tower (defined below) of a finite p-local spectrum



X


{\displaystyle X}

is



X


{\displaystyle X}

itself. The theorem was proved by Hopkins and Ravenel.




= Statement

=
Let




L

E
(
n
)




{\displaystyle L_{E(n)}}

denotes the Bousfield localization with respect to the Morava E-theory and let



X


{\displaystyle X}

be a finite,



p


{\displaystyle p}

-local spectrum. Then there is a tower associated to the localizations




⋯
→

L

E
(
2
)


X
→

L

E
(
1
)


X
→

L

E
(
0
)


X


{\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



X


{\displaystyle X}

.
The stages in the tower above are often simplifications of the original spectrum. For example,




L

E
(
0
)


X


{\displaystyle L_{E(0)}X}

is the rational localization and




L

E
(
1
)


X


{\displaystyle L_{E(1)}X}

is the localization with respect to p-local K-theory.



Stable homotopy groups

In particular, if the



p


{\displaystyle p}

-local spectrum



X


{\displaystyle X}

is the stable



p


{\displaystyle p}

-local sphere spectrum





S


(
p
)




{\displaystyle \mathbb {S} _{(p)}}

, then the homotopy limit of this sequence is the original



p


{\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.

Kata Kunci Pencarian:

• Chromatic homotopy theory
• Homotopy theory
• A¹ homotopy theory
• Morava K-theory
• Stable homotopy theory
• Rational homotopy theory
• Sphere spectrum
• Tomer Schlank
• Complex-oriented cohomology theory
• Redshift conjecture