nLab chromatic homotopy theory

Idea

Ravenel 1992, p. 24:

We use the word ‘chromatic’ here for the following reason. The $n$-th subquotients in the chromatic filtration consists of $v_n$-periodic elements. As illustrated in 2.4.2, these elements fall into periodic families. The chromatic filtration is thus like a spectrum in the astronomical sense in that it resolves the stable homotopy groups of a finite complex into periodic families of various periods. Comparing these to the colors of the rainbow led us to the word ‘chromatic’.

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 “chromatic” filtration 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 $p \in \mathbb{N}$ and for each $n \in \mathbb{N}$ there is a Bousfield localization of spectra

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

where $K(n)$ is the $n$th Morava K-theory (for the given prime $p$). 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 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 (see at smash product theorem and see this remark at fracture square )

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

it follows that in principle one may study a spectrum $X$ by understanding all its “chromatic pieces” $L_{K(n)} X$.

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

• motivation: J-homomorphism and chromatic homotopy

• stable homotopy theory

• stable homotopy groups of spheres, J-homomorphism

• complex oriented cohomology theory

• complex cobordism cohomology theory, MU

  • Lazard's theorem

  • Quillen's theorem on MU

  • Landweber exact functor theorem

  • Landweber-Novikov theorem

  • Brown-Peterson spectrum

• formal group law

  • moduli stack of formal groups

  • height of a formal group

  • Morava stabilizer group

• Morava K-theory

  • ∞-field

  • integral Morava K-theory

  • K(n)-local stable homotopy theory

  • Bousfield-Kuhn functor

• Morava E-theory

  • Lubin-Tate theory

• chromatic filtration

  • chromatic localization

  • chromatic tower

  • monochromatic layer

  • periodicity theorem

  • chromatic convergence theorem

• telescopic complexity?, telescopic localization

• Adams spectral sequence

  • Steenrod algebra, Adams operation

  • chromatic spectral sequence, May spectral sequence

• algebraic K-theory, red-shift conjecture

• transchromatic character

• motivic chromatic homotopy theory

References

Stable case

Original articles:

• Douglas Ravenel, Localization with Respect to Certain Periodic Homology Theories, American Journal of Mathematics Vol. 106, No. 2 (Apr., 1984), pp. 351-414 (doi:10.2307/2374308, jstor:2374308)

• Doug Ravenel, Nilpotence and Periodicity in Stable Homotopy Theory, Annals of Mathematics Studies 128, Princeton University Press (1992) [ISBN:9780691025728, pdf, webpage]

A quick idea:

• Mark Mahowald, Doug Ravenel, Section 6 of: Towards a Global Understanding of the Homotopy Groups of Spheres, pages 57-74 in Part II of: The Lefschetz Centennial Conference – Proceedings on Algebraic Topology, Proceedings of the Lefschetz Centennial Conference held December 10-14, 1984, Contemporary Mathematics 58, American Mathematical Society, 1987. (ISBN:978-0-8218-5063-3, pdf, pdf)

Historical introduction:

• Jack Morava, Complex cobordism and algebraic topology (arXiv:0707.3216)

Comprehensive lecture notes:

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010 (lecture notes)

• Doug Ravenel, Complex cobordism and stable homotopy groups of spheres

Survey:

• Haynes Miller, “Chromatic” homotopy theory May 2011 (pdf)

• Tobias Barthel, Agnès Beaudry, Chromatic structures in stable homotopy theory, in Handbook of Homotopy Theory, Chapman and Hall/CRC Press (2019) [arXiv:1901.09004, doi:10.1201/9781351251624]

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

• Hans-Werner Henn, Recent developments in chromatic stable homotopy theory [pdf&rbrack

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

• Talbot 2013: Chromatic Homotopy Theory, 2013 Pre-Talbot Seminar Chromatic homotopy theoy

See also:

• Glossary for stable and chromatic honotopy theory (pdf)

• David Mehrle, Chromatic homotopy theory: Journey to the frontier, Graduate workshop notes, Boulder 16-17 May 2018, (pdf, pdf)

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)

Relation of chromatic homotopy theory to Goodwillie calculus is discussed in

• Greg Arone, Mark Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Inventiones mathematicae, February 1999, Volume 135, Issue 3, pp 743-788 (pdf, doi:10.1007/s002220050300)

• Nicholas Kuhn, Goodwillie towers and chromatic homotopy: an overview, Proceedings of the Nishida Fest (Kinosaki 2003), 245–279, Geom. Topol. Monogr., 10, Geom. Topol. Publ., Coventry, 2007 (arXiv:math/0410342, doi:10.2140/gtm.2007.10.245)

Categorical ultraproducts are used to provide asymptotic approximations in

• Tobias Barthel, Tomer Schlank, Nathaniel Stapleton, Chromatic homotopy theory is asymptotically algebraic, (arXiv:1711.00844)

Unstable case

There are also chromatic approximations in ordinary (not stabilized) homotopy theory:

• Aldridge Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994), 831-873 (doi:10.1090/S0894-0347-1994-1257059-7, jstor:2152734)

• Gijs Heuts, Lie algebras and $v_n$-periodic spaces (arXiv:1803.06325)

• Jacob Lurie, Michael Hopkins, Unstable Chromatic Homotopy Theory, lecture notes 2018 (web)