Talk:Chromatic homotopy theory

Big Idea/Main Conjectures Needed
This article desperately needs to have the intuition behind chromatic homotopy theory, the construction of the chromatic tower, its main conjectors, and the big theorems such as the Chromatic Convergence Theorem. In addition, someone needs to figure out where to discuss $$v_k$$ periodicity. — Preceding unsigned comment added by 73.78.133.104 (talk) 20:40, 19 May 2017 (UTC)

Other resources
Wundzer (talk) 03:21, 21 June 2020 (UTC)
 * https://sites.math.northwestern.edu/~pgoerss/papers/modfg.pdf
 * https://etale.site/writing/chromatic-tower.pdf
 * https://web.math.rochester.edu/people/faculty/doug/mypapers/global.pdf
 * https://arxiv.org/abs/1901.07990
 * https://s.wayne.edu/isaksen/echt/
 * http://pi.math.cornell.edu/~dmehrle/notes/conferences/cht/cht-notes.pdf
 * https://web.math.rochester.edu/people/faculty/doug/otherpapers/coctalos.pdf

Breaking down Lurie's lectures
I'm using this a workspace for taking notes while writing up this page. Please do not delete!!!

Chromatic tower
Lecture 29



\cdots \to L_{E(2)}S_{(p)} \to L_{E(1)}S_{(p)} \to L_{E(0)}S_{(p)} $$

has inverse homotopy limit $$S_{(p)}$$. This is the chromatic convergence theorem


 * Chromatic convergence theorem/chromatic lower - lecture 29 - this depends on smashing localization and p-local spectra and Bousfield localization to p-local spectra
 * Need to have $$K(n)$$ and $$E(n)$$
 * https://ncatlab.org/nlab/show/smash+product+theorem — Preceding unsigned comment added by Wundzer (talk • contribs) 03:22, 21 June 2020 (UTC)

E(n) and K(n)
Lecture 22 Morava E-theory is an even periodic spectrum $$E(n)$$ whose homotopy groups $$\pi_*E(N)$$ are isomorphic to the ring $$R[\beta,\beta^{-1}]$$ where $$R$$ is the universal deformation ring of a formal group law of height $$n$$ over a field $$k$$, and $$\beta$$ is degree 2


 * Localization $$L_{E(n)}$$ is smashing, meaning it preserves direct sums
 * Define a spectrum $$M(k)$$ as a cofiber
 * Construct $$K(n)$$ from $$MU_{(p)}[\nu+p^{-1}]$$ and the $$M(k)$$ where $$k \neq p^n - 1$$

K(n) are "fields"
Lectures 23-25

Lecture titles
Wundzer (talk) 23:21, 20 June 2020 (UTC)
 * (1) Introduction
 * (2) Lazard's theorem
 * (3) Lazard's theorem continued
 * (4) Complex-Oriented Cohomology Theories
 * (5) Complex Bordism
 * (6) MU and Complex Orientations
 * (7) The homology of MU
 * (8) The Adams Spectral Sequence
 * (9) The Adams Spectral Sequence for MU
 * (10) The Proof of Quillen’s Theorem
 * (11) Formal Groups
 * (12) Heights of Formal Groups
 * (13) The Stratification of $$\mathcal{M}_{FG}$$
 * (14) Classification of Formal Groups
 * (15) Flat Modules over $$\mathcal{M}_{FG}$$
 * (16) The Landweber Exact Functor Theorem
 * (17) Phantom Maps
 * (18) Even Periodic Cohomology Theories
 * (19) Morava Stabilizer Groups
 * (20) Bousfield Localization
 * (21) Lubin-Tate Theory
 * (22) Morava E-Theory and Morava K-Theory
 * (23) The Bousfield Classes of E(n) and K(n)
 * (24) Uniqueness of Morava K-Theory
 * (25) The Nilpotence Theorem - useful for establishing stable homotopy groups are all nilpotent for postivie degrees
 * (26) Thick Subcategories - gives decomposition of stable homotopy category
 * (27) Periodicity theorem - technical theorem about p-local spectra
 * (28) Telescopic Localization
 * (29) Telescopic vs. $$E_n$$-Localization - Chromatic convergence theorem
 * (30) Localizations and the Adams-Novikov Spectral Sequence
 * (31) The Smash Product Theorem