nLab Chromatic Homotopy Theory

This page collects links related to the lecture notes

• Jacob Lurie,

  Chromatic Homotopy Theory

  Lecture series 2010

  web, pdf

on complex oriented cohomology, the Adams spectral sequence and chromatic homotopy theory from the modern point of view of E-infinity geometry.

Based on the program of

• Frank Adams, Stable homotopy and generalised homology, 1974

as nicely laid out in more detail in

• Stanley Kochmann, Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

First indications of the big picture developed here are due to

• Mike Hopkins, Complex oriented cohomology theories and the language of stacks, 1999

Via the chromatic stratification of the moduli stack of formal groups one recovers as, roughly, the second chromatic stage the moduli stack of elliptic curves. For the story in that case see

• Jacob Lurie, A Survey of Elliptic Cohomology.

See also

• Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, 1986/2003

Lectures

• Lecture 1 Introduction (pdf)

• Lecture 2 Lazard's theorem (pdf)

• Lecture 3 Lazard’s theorem (continued) (pdf)

• Lecture 4 Complex-oriented cohomology theories (pdf)

• Lecture 5 Complex bordism (pdf)

• Lecture 6 MU and complex orientations (pdf)

• Lecture 7 The homology of MU (pdf)

• Lecture 8 The Adams spectral sequence (pdf)

• Lecture 9 The Adams spectral sequence for MU (pdf)

• Lecture 10 The proof of Quillen's theorem (pdf)

• Lecture 11 Formal groups (pdf)

• Lecture 12 Heights and formal groups (pdf)

• Lecture 13 The stratification of $\mathcal{M}_{FG}$ (pdf)

• Lecture 14 Classification of formal groups (pdf)

• Lecture 15 Flat modules over $\mathcal{M}_{FG}$ (pdf)

• Lecture 16 The Landweber exact functor theorem (pdf)

• Lecture 17 Phantom maps (pdf)

• Lecture 18 Even periodic cohomology theories (pdf)

• Lecture 19 Morava stabilizer groups (pdf)

• Lecture 20 Bousfield localization (pdf)

• Lecture 21 Lubin-Tate theory (pdf)

• Lecture 22 Morava E-theory and Morava K-theory (pdf)

• Lecture 23 The Bousfield Classes of $E(n)$ and $K(n)$ (pdf)

• Lecture 24 Uniqueness of Morava K-theory (pdf)

• Lecture 25 The Nilpotence lemma (pdf)

• Lecture 26 Thick subcategories (pdf)

• Lecture 27 The periodicity theorem (pdf)

• Lecture 28 Telescopic localization (pdf)

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

• Lecture 30 Localizations and the Adams-Novikov spectral sequence (pdf)

• Lecture 31 The smash product theorem (pdf)

• Lecture 32 The chromatic convergence theorem (pdf)

• Lecture 33 Complex bordism and $E(n)$-localization (pdf)

• Lecture 34 Monochromatic layers (pdf)

• Lecture 35 The image of $J$ (pdf)

Lecture 1 Introduction

Lecture 2 Lazard’s theorem

• Lazard ring

• Lazard's theorem

Lecture 3 Lazard’s theorem (continued)

• Lazard's theorem

Lecture 4 Complex-oriented cohomology theories

• complex oriented cohomology theory

• splitting principle

• generalized Chern classes

Lecture 5 Complex bordism

• complex cobordism cohomology theory, MU

Lecture 6 MU and complex orientations

• universal complex orientation on MU

Lecture 7 The homology of MU

• Boardman homomorphism

• homology of MU

Lecture 8 The Adams spectral sequence

• Adams resolution

• Adams spectral sequence

Lecture 9 The Adams spectral sequence for MU

• Adams-Novikov spectral sequence

Lecture 10 The proof of Quillen’s theorem

• Quillen's theorem on MU

Lecture 11 Formal groups

• formal group

Lecture 12 Heights and formal groups

• height of a formal group

Lecture 13 The stratification of $\mathcal{M}_{FG}$

• chromatic filtration

Lecture 14 Classification of formal groups

• moduli stack of formal groups

Lecture 15 Flat modules over $\mathcal{M}_{FG}$

• quasicoherent sheaf, flat module, exact functor

• Landweber exactness

Lecture 16 The Landweber exact functor theorem

• Landweber exact functor theorem

Lecture 17 Phantom maps

• Brown representability theorem, Landweber exact spectrum

• phantom map

Lecture 18 Even periodic cohomology theories

• periodic cohomology theory

• periodic ring spectrum

Lecture 19 Morava stabilizer groups

• Morava stabilizer group

Lecture 20 Bousfield localization

• Bousfield localization

• Bousfield localization of spectra

• p-localization

Lecture 21 Lubin-Tate theory

• Lubin-Tate formal group

• Lubin-Tate theory

Lecture 22 Morava E-theory and Morava K-theory

• Morava K-theory

• Morava E-theory

Lecture 23 The Bousfield Classes of $E(n)$ and $K(n)$

• Bousfield equivalence

• smash product theorem

Lecture 24 Uniqueness of Morava K-theory

• ∞-field

Lecture 25 The Nilpotence lemma

• nilpotence lemma

Lecture 26 Thick subcategories

• thick subcategory

Lecture 27 The periodicity theorem

• periodicity theorem

Lecture 28 Telescopic localization

• telescopic localization

Lecture 29 Telescopic vs $E_n$-localization

• smashing localization

• chromatic tower

Lecture 30 Localizations and the Adams-Novikov spectral sequence

• Adams-Novikov spectral sequence

Lecture 31 The smash product theorem

• smash product theorem

Lecture 32 The chromatic convergence theorem

• chromatic convergence theorem

Lecture 33 Complex bordism and $E(n)$-localization

Lecture 34 Monochromatic layers

• monochromatic layer

Lecture 35 The image of $J$

• J-homomorphism

• Adams operation

category: reference