nLab
chromatic homotopy theory

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Higher algebra

Contents

Idea

By the construction of complex oriented cohomology theories from formal groups (via the Landweber exact functor theorem), the height filtration of formal groups induces a “chromaticfiltration on complex oriented cohomology theories. Chromatic homotopy theory is the study of stable homotopy theory and specifically of complex oriented cohomology theories by means of and along this chromatic filtration.

More abstractly, this filtering is induced by the prime spectrum of a symmetric monoidal stable (∞,1)-category of the (∞,1)-category of spectra for p-local finite spectra, which is labeled by the Morava K-theories.

More in detail, for each prime pp \in \mathbb{N} and for each nn \in \mathbb{N} there is a Bousfield localization of spectra

L nL K(0)K(n), L_n \coloneqq L_{K(0)\vee \cdots \vee K(n)} \,,

where K(n)K(n) is the nnth Morava K-theory (for the given prime pp). These arrange into the chromatic tower which for each spectrum XX is of the form

XL nXL n1XL 0X. X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X \,.

The chromatic convergence theorem states mild conditions under which the homotopy limit over this tower is the pp-localization

XX (p) X \to X_{(p)}

of XX.

Since moreover L nXL_n X is the homotopy fiber product

L nXL K(n)X×L n1L K(n)XL n1X L_n X \simeq L_{K(n)}X \underset{L_{n-1}L_{K(n)}X}{\times} L_{n-1}X

it follows that in principle one can study a spectrum XX by understanding all its “chromatic pieces” L K(n)XL_{K(n)} X.

Examples

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory

References

A quick idea is given in section 6 of

A good historical introduction is in

Comprehensive lecture notes are in

Brief surveys include

A lightning review of results by Henn with Goerss, Mahowald, Rezk, and Karamanov is in

  • Hans-Werner Henn, Recent developments in stable homotopy theory (pdf)

A collection of various lecture notes and other material is kept at

Random useful discussion is in

Discussion of generalization of elliptic cohomology to higher chromatic homotopy theory is discussed in

  • Doug Ravenel, Toward higher chromatic analogs of elliptic cohomology pdf

  • Doug Ravenel, Toward higher chromatic analogs of elliptic cohomology II, Homology, Homotopy and Applications, vol. 10(1), 2008, pp.1-36 (pdf, pdf slides)

Revised on May 2, 2014 05:08:36 by Urs Schreiber (89.204.137.81)