# Chromatic Homotopy Theory

This is a joint seminar of the Center for Advanced Studies at Skoltech and International laboratory for Mirror Symmetry and Automorphic Forms offered by myself and Vladimir Shaydurov in the 2018-2019 academic year. The lectures are scheduled on Tuesday, 13:00; room 335 (Skoltech: 1 Nobel Street, red building; public transport).

## Description

One important problem of the modern algebraic topology is to understand the structure of the ring $\pi_*(\mathbb S)$ of stable homotopy groups of spheres. There are two big classical results in this direction

• Nishida's theorem asserts that every class in $\pi_{>0}(\mathbb S)$ is nilpotent.
• Adams' theorem about the image of $J$-homomorphism provides us with the infinite family of elements in $\pi_*(\mathbb S)$.

It turns out these results of seemingly distinct nature are in fact related to each other and both find natural generalizations in the branch of algebraic topology called chromatic homotopy theory. Namely by contrast to the sphere spectrum case, for each odd prime number $p$ Adams constructed a self-map $v_1\colon \Sigma^{2(p-1)}\mathbb S/p \to \mathbb S/p$ which induces multiplication by $\beta^{p-1}$ in $K$-theory and hence can not be nilpotent (the story for $p=2$ works similarly). The $p$-torsion of the image of $J$-homomorphism then can be obtained from iterates of $v_1$ by taking the composition $\mathbb S^{2(p-1)i} \to \mathbb S^{2(p-1)i}/p \stackrel{v_1^i}{\longrightarrow} \mathbb S/p \to \mathbb S^1$. The natural question now is to try to generalize this picture, e.g. can we find a non-nilpotent self-map $v_2\colon \Sigma^r \mathbb S/(p,v_1) \to \mathbb S/(p,v_1)$? To answer such questions it is desirable to have some control over nilpotency. Very satisfactory solution is provided by the following celebrated theorem of Devinatz-Hopkins-Smith:

• (Nilpotence theorem). Let $R$ be a homotopy-associative ring spectrum. Then the element $x\in \pi_* R$ is nilpotent if and only if its image in $MU_*R$ under the Hurewicz map is nilpotent.
• Note that the Nishida's theorem is an easy consequence, since $\pi_{>0}(\mathbb S)$ is torsion and $\pi_*(MU)$ is torsion free.

To explain the power of the nilpotence theorem

, we need first to discuss the geometry of the complex cobordism spectrum $MU$. By the theorems of Miller-Quillen and Landweber-Novikov the Hopf algebroid $(MU_*, MU_*MU)$ corepresents the stack $\mathcal M_{fg}$ of $1$-dimensional commutative formal group laws. Now $p$-locally every formal group has an invariant called height: the height of formal multiplicative group is $1$, all formal groups of heights $\le 2$ can be obtained from elliptic curves, with formal groups of heights $2$ corresponding precisely to supersingular elliptic curves, e.t.c. It follows the stack $\mathcal M_{fg,(p)}:= \mathcal M_{fg}\otimes_{\mathbb Z} \mathbb Z_{(p)}$ admits an increasing filtration by open substacks $\mathcal M_{fg,(p)}^{\le h}$ of formal group laws of heights less then $h+1$ with locally closed strata $\mathcal M_{fg,(p)}^h$ being substacks of formal group laws of heights exactly $h$. Given any spectrum $X$ its $MU$-homology $MU_*(X)$ are naturally co-module over $(MU_*, MU_*MU)$, hence descent to give a quasi-coherent sheaf $\mathcal F_X\in \mathrm{QCoh}(M_{fg})$ (in particular the second page of $MU$-based Adams-Novikov spectral sequence for $X$ can be identified with $H^*(\mathcal M_{fg}, \mathcal F_X)$). Given any sheaf $\mathcal F\in \mathrm{QCoh}(\mathcal M_{fg,(p)})$ one can restrict $\mathcal F$ to an open substack $\mathcal M_{fg,(p)}^{\le h}$ or take its completion in $\mathcal M_{fg,(p)}^{h}$. It turns out these operations can be lifted to the category of spectra: for each prime $p$ and natural $n$ there exist spectra $E(n)$ and $K(n)$ (Morava's $E$- and $K$-theories) such that Bousfield localizations $L_{E(n)}$ and $L_{K(n)}$ correspond to restriction to $\mathcal M_{fg,(p)}^{\le h}$ and completion in $\mathcal M_{fg,(p)}^{h}$ respectively (e.g. $L_{E(n)} X \simeq 0$ if and only if the restriction of the associated sheaf $\mathcal F_X$ to $\mathcal M_{fg,(p)}^{> h}$ vanish). For $n=0$ we have $E(0) \simeq K(0) \simeq H\mathbb Q_p$, for $n=1$ there is a splitting $KU/p \simeq \bigoplus_{i=0}^{p-1} \Sigma^{2i} K(1)$, hence the name. In general $\pi_* E(n) \simeq \mathbb Z_p[[v_1,\ldots, v_{n-1}]][\beta^{\pm 1}]$ and $\pi_* K(n) \simeq \mathbb F_p[v_n^{\pm 1}]$ with $|v_i| = 2(p^i -1), |\beta|=2$.

Since $L_{E(n-1)} \circ L_{E(n)} \simeq L_{E(n-1)}$, for any spectrum $X$ we can then consider its chromatic tower $X \to \ldots \to L_{E(n)} X \to L_{E(n-1)} X \to \ldots \to L_{E(0)} X \simeq X\wedge H\mathbb Q_p$. One important consequence of the nilpotence theorem asserts

• (Chromatic convergence theorem). Let $X$ be a finite $p$-local spectrum. Then the canonical map $X \to \lim\limits_{\leftarrow} L_{E(n)} X$ is an equivalence.
• Now if we denote by $M_nX$ the fiber of $L_{E(n)}X \to L_{E(n-1)}$, by standard machinery there is a spectral sequence starting from $\pi_*(M_n X)$ and converging to $\pi_*(X)$. To understand the monochromatic layer $M_n X$ note that $\mathcal M_{fg,(p)}^{\le h} \simeq \mathcal M_{fg,(p)}^{\le h-1} \bigcup_{\mathcal M_{fg,(p)}^{\widehat h, \circ}} \mathcal M_{fg,(p)}^{\widehat h}$, where $\mathcal M_{fg,(p)}^{\widehat h}$ stands for completion of $\mathcal M_{fg,(p)}^{\le h}$ in $\mathcal M_{fg,(p)}^h$ and $\mathcal M_{fg,(p)}^{\widehat h, \circ} := \mathcal M_{fg,(p)}^{\widehat h}\bigcap \mathcal M_{fg,(p)}^{\le h-1}$ is the punctured neighborhood. It follows by Mayer-Vietoris sequence that for $\mathcal F \in \mathrm{QCoh}(\mathcal M_{fg,(p)}^{\le h})$ the fiber of $\mathcal F \to \mathcal F_{|\mathcal M_{fg,(p)}^{\le h - 1}}$ is equivalent to the fiber of $\mathcal F_{\widehat{\mathcal M_{fg,(p)}^h}} \to (\mathcal F_{\widehat{\mathcal M_{fg,(p)}^h}})_{|\mathcal M_{fg,(p)}^{\le h - 1}}$. Completely analogue statement holds in spectra, i.e. $M_n X \simeq \mathrm{fib}(L_{K(n)} X \to L_{E(n-1)} L_{K(n)} X)$. This is useful because we understand $L_{K(n)}$ since the stack $\mathcal M_{fg,(p)}^{h}$ is tractable: over an algebraicly closed field there exists a unique up to isomorphism formal group law of heights exactly $h$ (let us denote it by $f_h$), hence $\mathcal M_{fg,(p)}^{h} \simeq B\mathbb G_h$ where $\mathbb G_h$ is the group of automorphism of $f_h$, the so called Morava stabilizer group. Motivated by this Davinantz-Hopkins proved that there is an action of $\mathbb G_n$ on $E(n)$ and a natural equivalence $L_{K(n)} X \simeq E(n)^{h\mathbb G_n}\wedge X$ for finite $X$.

Back to $v_n$-self maps let us say that $X$ has type $n$ if $L_{E(n-1)} X \simeq 0$ and $L_{E(n)} X \not \simeq 0$. Nilpotence theorem also serves as an important input to the proof of the following result

• (Periodicity theorem). Let $X$ be a finite $p$-local spectrum of type $n$. Then $X$ admits a (essentially unique) $v_n$-self map, i.e. a map $f\colon \Sigma^r X \to X$ which induces multiplication by some power of $v_n$ in $K(n)_*(X)$.
• By construction the spectrum $M_n X$ is of type $n$ but rarely finite. Nevertheless one can prove that $M_n X$ is a filtered colimit of finite spectra of type $n$. It follows every element of homotopy groups $\pi_* M_n X$ lives in a periodic family. Applying these to $X=\mathbb S$ one finds that every class in $\pi_* M_n X$ factors through the iterated Moore spectrum $\mathbb S/(p^{i_0}, v_1^{i_1}, \ldots, v_{n-1}^{i_{n-1}})$ for some $i_0, i_1, \ldots, i_{n-1}$ and admits a $v_n$-self map $v_n^{i_n}$. By taking the composition $\alpha_{\overline i}\colon \mathbb S^r \to \mathbb S^r/(p^{i_0}, v_1^{i_1}, \ldots, v_{n-1}^{i_{n-1}}) \stackrel{v_n^{j_n}}{\longrightarrow} \mathbb S/(p^{i_0}, v_1^{i_1}, \ldots, v_{n-1}^{i_{n-1}}) \to \mathbb S^t$ (where the first map is an inclusion of the bottom cell and the last map is the projection on the top cell) we obtain the so-called Greek letter element. Note that by chromatic convergence any element of the stable homotopy groups of spheres can be obtained this way.

### Plan

1. Reminder on spectra and Bousfield localizations.
2. Complex oriented cohomology theories. Miller-Quillen's theorem. Moduli stack of formal groups. Landweber exact functor theorem. Morava $E$- and $K$-theories.
3. $E(n)$- and $K(n)$-local categories. Morava stabilizer group. $v_n$-self maps.
4. Nilpotence theorem. Periodicity theorem. Thick subcategory theorem. Smash product theorem. Chromatic convergence theorem. Fracture squares.
5. Hopkins-Devinatz theorem. Galois theory of Rognes-Mathew.
6. Shimura varieties and automorphic forms.
7. Elements of spectral algebraic geometry. Artin's representability theorem. Goerss-Hopkins-Miller-Lurie theorem. Topological automorphic forms.
8. Finite resolutions of K(n)-local sphere spectra and congruences in rings of automorphic forms.

## Talks

Talk Speaker Date, place
Overview of the courseArtem PrikhodkoSeptember, 18
Stable categories and spectra IArtem PrikhodkoSeptember, 25
Introduction to higher categoriesArtem PrikhodkoSeptember, 30
Presentable categories IArtem PrikhodkoOctober, 2
Presentable categories II
Stable categories and spectra II
Artem PrikhodkoOctober, 7
Stable categories and spectra IIIArtem PrikhodkoOctober, 9
Bousfield localisationsArtem PrikhodkoOctober, 13
Complex orientable cohomology theoriesArtem PrikhodkoOctober, 16
Steenrod algebraVladimir ShaydurovOctober, 20
Canceled due to Recent developments in Higgs theory conferenceOctober, 23
Exercise sessionArtem PrikhodkoOctober, 27
Adams-Novikov spectral sequence IArtem PrikhodkoOctober, 30
Adams-Novikov spectral sequence IIArtem PrikhodkoNovember, 3
Thom spectra, MUArtem PrikhodkoNovember, 6
The image of $J$-homomorphism IVladimir ShaydurovNovember, 10
Moduli stacks of formal groupsKirill SalmagambetovNovember, 13
The image of $J$-homomorphism IIVladimir ShaydurovNovember, 17
Lazard's theoremAlexander NovikovNovember, 20
Quillen's theoremAlexander NovikovNovember, 24
Landweber's criterionDmitrii ZhurbenkoNovember, 27
Lubin-Tate FGL, Morava $E$-theoriesArtem PrikhodkoDecember, 1
Morava $K$-theoriesArtem PrikhodkoDecember, 4
Lazard's open subgroup theoremDmitrii KrekovDecember, 8
On Borel's equivariant cohomology IVasya RogovDecember, 11
Morava stabilizer group and its cohomologyRoman KositsinDecember, 15
On Borel's equivariant cohomology IIVasya RogovDecember, 18
Nilpotence theorem: implications (thick subcategory theorem, ...)Pavel PopovDecember, 22
Nilpotence theorem: proof IArtem PrikhodkoDecember, 25
Nilpotence theorem: proof IIArtem PrikhodkoDecember, 29
$v_n$-self maps, periodicity theorem IEgor ShuklinJanuary, 19
Periodicity theorem IIEgor ShuklinJanuary, 22
Periodicity theorem IIIVladimir ShaydurovJanuary, 26
Monochromatic categoriesArtem PrikhodkoFebruary, 2
Chromatic convergecnce theoremArtem PrikhodkoFebruary, 9
Smash localization theoremArtem PrikhodkoFebruary, 12
Overview of the second part of the courseArtem PrikhodkoFebruary, 16
Introduction to operads and monadsArtem PrikhodkoFebruary, 19
Examples of operadsArtem PrikhodkoFebruary, 23
May recognitionArtem PrikhodkoFebruary, 26
Algebraic cobordism: Riemann-Roch theorems, operations and structurePavel SechinMarch, 2
Aside: (semi-)topological $K$-theoryAndrei KonovalovMarch, 9
Goerss-Hopkins obstruction theory IArtem PrikhodkoMarch, 12
Locally symmetric spaces and aproximation theoremsPavel PopovMarch, 19
Goerss-Hopkins obstruction theory IIArtem PrikhodkoMarch, 23
Goerss-Hopkins obstruction theory IIIArtem PrikhodkoMarch, 27
Introduction to Shimura varietie IRoman KositsinMarch, 30
Introduction to Shimura varietie IIRoman KositsinMarch, 2
Introduction to Shimura varietie IIIRoman KositsinMarch, 6
Cotangent complex IEgor ShuklinApril, 13
Cotangent complex IIEgor ShuklinApril, 20
Deformations of commutative algebras IArtem PrikhodkoApril, 23
Deformations of commutative algebras II
Reminder of DAG
Artem PrikhodkoApril, 30
Artin's representability theoremAlexander NovikovMay, 7
Serre-Tate theoryIvan PerunovMay, 14
Galois theory and homotopy theoryCatherine RayAugust, 15

Some propblems: Problem set 1a, Problem set 1b, Problem set 4.

## References

### Chromatic homotopy

1. Jacob Lurie, Chromatic Homotopy Theory.
2. Michael Hopkins. Complex Oriented Cohomology Theories and the Language of Stacks.
3. Douglas Ravenel. Complex Cobordism and Stable Homotopy Groups of Spheres (green book).
4. Douglas Ravenel. Nilpotence and periodicity in stable homotopy theory (orange book).
5. Douglas Ravenel. Localization with respect to certain periodic homology theories.
6. Ethan Devinatz, Michael Hopkins, Jeffrey Smith. Nilpotence and Stable Homotopy Theory I, II.
7. Aldridge Bousfield. The localization of spectra with respect to homology.
8. John Rognes, The Adams Spectral Sequence.

### Shimura varieties and automorphic forms

9. Pierre Deligne, Travaux de Shimura.
10. Alain Genestier, Bao Chau Ngo, Lectures on Shimura varieties.
11. Armand Borel, Introduction to the Cohomology of Arithmetic Groups.
12. Gunter Harder, Cohomology of Arithmetic Groups.
13. Stephen Gelbart, Automorphic Forms on Adele Groups.

### Topological modular and automorphic forms

14. Jacob Lurie, A survey of elliptic cohomology.
15. Jacob Lurie, Elliptic Cohomology I, II.
16. Mark Behrens, A modular description of the K(2)-local sphere at the prime 3.
17. Mark Behrens, Tyler Lawson, Topological Automorphic Forms.
18. Mark Behrens, Topological modular and automorphic forms.

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