Chromatic spectral sequence

Chromatic spectral sequence
Name	Chromatic spectral sequence
Field	Algebraic topology
Introduced	1970s
Notable	Douglas Ravenel, Michael Hopkins, Haynes Miller, Mark Mahowald
Related	Adams–Novikov spectral sequence, Morava K-theory, Brown–Peterson cohomology

Chromatic spectral sequence The chromatic spectral sequence is a computational tool in Algebraic topology and Stable homotopy theory that organizes information about the stable homotopy groups of spheres using the chromatic filtration and height stratification associated to formal group laws and complex cobordism. It arises from work of Douglas C. Ravenel, Haynes Miller, Mark Hovey, Charles Rezk, and Michael J. Hopkins and connects to major structures such as the Adams–Novikov spectral sequence, Brown–Peterson cohomology, and Morava K-theory. The spectral sequence encodes interactions among periodic families discovered in studies by John Milnor, Frank Adams, Adolf Hurwitz, and later developments in elliptic cohomology and topological modular forms.

Introduction

The chromatic spectral sequence refines the approach of the Adams spectral sequence and the Adams–Novikov spectral sequence by filtering spectra through a chromatic tower inspired by the work of Jean-Pierre Serre, Alexander Grothendieck, and Michel Demazure on formal groups. It leverages apparatus from Brown–Peterson cohomology and the theory of Landweber exact functor theorem developed by Peter Landweber and relates to duality phenomena studied by J. Peter May and G. W. Whitehead. The conceptual framework ties into breakthroughs by Ravenel summarized in conjectures bearing his name and validated in part by work of Devinatz, Hopkins, and Smith.

Construction and definition

The construction begins with the chromatic filtration of the sphere spectrum produced via Bousfield localization at Morava E-theory and Morava K-theory; central contributors include Jack Morava and Eduard Looijenga. One forms a tower of localized spectra using idempotent localizations akin to methods of Daniel Quillen and Benson Farb and then applies the Adams–Novikov spectral sequence machinery to the successive fibers, echoing techniques of J. F. Adams and Douglas Ravenel. The E_1-page is assembled from Ext-groups computed in categories related to BP_*BP comodules and uses input from the work of M. Hazewinkel on formal groups; later pages reflect differentials connecting heights via operations reminiscent of those in Hopf algebras studied by Milnor.

Relation to chromatic filtration and Morava K-theories

The spectral sequence organizes contributions of height-n phenomena detected by Morava K-theory K(n) and completed theories such as Morava E-theory E_n, building on insights from Jack Morava and Haynes Miller. It implements the chromatic filtration central to Ravenel's conjectures and interfaces with the thick subcategory theorem proven by Devinatz, Hopkins, and Smith. Connections run through formal group law invariants central to Lubin–Tate theory and use techniques from local field theory and p-adic Hodge theory pioneered by John Tate and Jean-Marc Fontaine in related contexts of deformation spaces considered by Mazur and Drinfeld.

Computations and examples

Explicit computations use the algebraic input from Brown–Peterson cohomology BP and the structure of BP_*BP comodules studied by William G. Dwyer and Daniel Kan. Classical computations at low chromatic heights involve families discovered by Mark Mahowald and J. F. Adams such as the image of J, Greek letter elements, and periodic v_n-towers analyzed by Haynes Miller and Douglas Ravenel. Higher-height calculations exploit techniques from elliptic cohomology and topological modular forms (tmf) developed by Hopkins, Haynes Miller, Mark Behrens, and Matthew Ando; sample computations reference work on the stable homotopy groups of spheres up to chromatic heights studied by Isaksen and Hill Hopkins Ravenel.

Convergence and collapse results

Convergence questions tie to chromatic convergence theorems proved by Ravenel and refinements due to Devinatz and Hopkins; partial collapse results appear in contexts where vanishing lines or sparseness conditions occur, reminiscent of collapse phenomena in the Adams spectral sequence studied by Frank Adams and Douglas Ravenel. Results proving strong convergence in localized or periodic settings use structural theorems from model category theory of Quillen and rely on input from the nilpotence and periodicity theorems by Devinatz, Hopkins, and Smith.

Applications and significance in stable homotopy theory

The chromatic spectral sequence provides a framework for organizing and predicting periodic phenomena in the stable homotopy groups of spheres and informs constructions in topological modular forms, tmf, and equivariant versions developed by Mike Hill, Michael Hopkins, and Doug Ravenel. It influences computations in cobordism theory initiated by René Thom and connects to geometric applications involving manifold invariants and structured ring spectra studied by Elmendorf, Kriz, and Mandell. The perspective has stimulated cross-fertilization with arithmetic geometry via links to Lubin–Tate spaces, moduli of formal groups, and deformation theory as developed by Deligne and Drinfeld, shaping current research directions in chromatic homotopy theory and related areas explored at institutions such as Institute for Advanced Study and Mathematical Sciences Research Institute.

Category:Algebraic topology