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chromatic tower

Contents

Context

Stable Homotopy theory

Higher algebra

Contents

Idea

For each prime pp \in \mathbb{N} and for each natural number nn \in \mathbb{N} there is a Bousfield localization of spectra

L nL E(n), L_n \coloneqq L_{E(n)} \,,

where E(n)E(n) is the nnth Morava E-theory (for the given prime pp), the nnth chromatic localization. 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 homotopy fibers of each stage of the tower

M n(X)fib(L E(n)XL E(n1)(X)) M_n(X) \coloneqq fib(L_{E(n)}X \longrightarrow L_{E(n-1)}(X))

is called the nnth monochromatic layer of XX.

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

(see at smash product theorem) it follows that in principle one can study a spectrum XX by understanding all its “chromatic pieces” L K(n)XL_{K(n)} X. This is the topic of chromatic homotopy theory.

Properties

Chromatic spectral sequence

The spectral sequence of a filtered stable homotopy type associated with the chromatic tower (regarded as a filtered object in an (infinity,1)-category) is the chromatic spectral sequence (Wilson 13, section 2.1.2)

tower diagram/filteringspectral sequence of a filtered stable homotopy type
filtered chain complexspectral sequence of a filtered complex
Postnikov towerAtiyah-Hirzebruch spectral sequence
chromatic towerchromatic spectral sequence
skeleta of simplicial objectspectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebraAdams spectral sequence
filtration by support
slice filtrationslice spectral sequence

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

Last revised on November 18, 2013 at 21:18:42. See the history of this page for a list of all contributions to it.