nLab chromatic tower

Idea

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

$$L_n \coloneqq L_{E(n)} \,,$$

where $E(n)$ is the $n$th Morava E-theory (for the given prime $p$), the $n$th chromatic localization. These arrange into the chromatic tower which for each spectrum $X$ is of the form

$$X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X \,.$$

The homotopy fibers of each stage of the tower

$$M_n(X) \coloneqq fib(L_{E(n)}X \longrightarrow L_{E(n-1)}(X))$$

is called the $n$th monochromatic layer of $X$.

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

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

of $X$.

Since moreover $L_n X$ is the homotopy fiber product

$$L_n X \simeq L_{K(n)}X \underset{L_{n-1}L_{K(n)}X}{\times} L_{n-1}X$$

(see at smash product theorem) it follows that in principle one can study a spectrum $X$ by understanding all its “chromatic pieces” $L_{K(n)} X$. 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/filtering	spectral sequence of a filtered stable homotopy type
filtered chain complex	spectral sequence of a filtered complex
Postnikov tower	Atiyah-Hirzebruch spectral sequence
chromatic tower	chromatic spectral sequence
skeleta of simplicial object	spectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebra	Adams spectral sequence
filtration by support	…
slice filtration	slice spectral sequence

Examples

chromatic homotopy theory

chromatic level	complex oriented cohomology theory	E-∞ ring/A-∞ ring	real oriented cohomology theory
0	ordinary cohomology	Eilenberg-MacLane spectrum $H \mathbb{Z}$	HZR-theory
	0th Morava K-theory	$K(0)$
1	complex K-theory	complex K-theory spectrum $KU$	KR-theory
	first Morava K-theory	$K(1)$
	first Morava E-theory	$E(1)$
2	elliptic cohomology	elliptic spectrum $Ell_E$
	second Morava K-theory	$K(2)$
	second Morava E-theory	$E(2)$
	algebraic K-theory of KU	$K(KU)$
3 …10	K3 cohomology	K3 spectrum
$n$	$n$th Morava K-theory	$K(n)$
	$n$th Morava E-theory	$E(n)$	BPR-theory
$n+1$	algebraic K-theory applied to chrom. level $n$	$K(E_n)$ (red-shift conjecture)
$\infty$	complex cobordism cohomology	MU	MR-theory

Related concepts

• Postnikov tower, Taylor tower

References

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010,

  Lecture 29 Telescopic vs $E_n$-localization (pdf)

• Dylan WilsonSpectral Sequences from Sequences of Spectra: Towards the

  Spectrum of the Category of Spectra_ lecture at 2013 Pre-Talbot Seminar (pdf)