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
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:
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
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
Talk | Speaker | Date, place |
---|---|---|
Overview of the course | Artem Prikhodko | September, 18 |
Stable categories and spectra I | Artem Prikhodko | September, 25 |
Introduction to higher categories | Artem Prikhodko | September, 30 |
Presentable categories I | Artem Prikhodko | October, 2 |
Presentable categories II Stable categories and spectra II | Artem Prikhodko | October, 7 |
Stable categories and spectra III | Artem Prikhodko | October, 9 |
Bousfield localisations | Artem Prikhodko | October, 13 |
Complex orientable cohomology theories | Artem Prikhodko | October, 16 |
Steenrod algebra | Vladimir Shaydurov | October, 20 |
Canceled due to Recent developments in Higgs theory conference | October, 23 | |
Exercise session | Artem Prikhodko | October, 27 |
Adams-Novikov spectral sequence I | Artem Prikhodko | October, 30 |
Adams-Novikov spectral sequence II | Artem Prikhodko | November, 3 |
Thom spectra, MU | Artem Prikhodko | November, 6 |
The image of $J$-homomorphism I | Vladimir Shaydurov | November, 10 |
Moduli stacks of formal groups | Kirill Salmagambetov | November, 13 |
The image of $J$-homomorphism II | Vladimir Shaydurov | November, 17 |
Lazard's theorem | Alexander Novikov | November, 20 |
Quillen's theorem | Alexander Novikov | November, 24 |
Landweber's criterion | Dmitrii Zhurbenko | November, 27 |
Lubin-Tate FGL, Morava $E$-theories | Artem Prikhodko | December, 1 |
Morava $K$-theories | Artem Prikhodko | December, 4 |
Lazard's open subgroup theorem | Dmitrii Krekov | December, 8 |
On Borel's equivariant cohomology I | Vasya Rogov | December, 11 |
Morava stabilizer group and its cohomology | Roman Kositsin | December, 15 |
On Borel's equivariant cohomology II | Vasya Rogov | December, 18 |
Nilpotence theorem: implications (thick subcategory theorem, ...) | Pavel Popov | December, 22 |
Nilpotence theorem: proof I | Artem Prikhodko | December, 25 |
Nilpotence theorem: proof II | Artem Prikhodko | December, 29 |
$v_n$-self maps, periodicity theorem I | Egor Shuklin | January, 19 |
Periodicity theorem II | Egor Shuklin | January, 22 |
Periodicity theorem III | Vladimir Shaydurov | January, 26 |
Monochromatic categories | Artem Prikhodko | February, 2 |
Chromatic convergecnce theorem | Artem Prikhodko | February, 9 |
Smash localization theorem | Artem Prikhodko | February, 12 |
Overview of the second part of the course | Artem Prikhodko | February, 16 |
Introduction to operads and monads | Artem Prikhodko | February, 19 |
Examples of operads | Artem Prikhodko | February, 23 |
May recognition | Artem Prikhodko | February, 26 |
Algebraic cobordism: Riemann-Roch theorems, operations and structure | Pavel Sechin | March, 2 |
Aside: (semi-)topological $K$-theory | Andrei Konovalov | March, 9 |
Goerss-Hopkins obstruction theory I | Artem Prikhodko | March, 12 |
Locally symmetric spaces and aproximation theorems | Pavel Popov | March, 19 |
Goerss-Hopkins obstruction theory II | Artem Prikhodko | March, 23 |
Goerss-Hopkins obstruction theory III | Artem Prikhodko | March, 27 |
Introduction to Shimura varietie I | Roman Kositsin | March, 30 |
Introduction to Shimura varietie II | Roman Kositsin | March, 2 |
Introduction to Shimura varietie III | Roman Kositsin | March, 6 |
Cotangent complex I | Egor Shuklin | April, 13 |
Cotangent complex II | Egor Shuklin | April, 20 |
Deformations of commutative algebras I | Artem Prikhodko | April, 23 |
Deformations of commutative algebras II Reminder of DAG | Artem Prikhodko | April, 30 |
Artin's representability theorem | Alexander Novikov | May, 7 |
Serre-Tate theory | Ivan Perunov | May, 14 |
Galois theory and homotopy theory | Catherine Ray | August, 15 |
Some propblems: Problem set 1a, Problem set 1b, Problem set 4.