# nLab chromatic tower

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

For each prime $p \in \mathbb{N}$ and for each natural number $n \in \mathbb{N}$ there is a Bousfield localization of spectra

$L_n \coloneqq L_{E(n)} \,,$

where $E(n)$ is the $n$th Morava E-theory (for the given prime $p$), the $n$th chromatic localization. These arrange into the chromatic tower which for each spectrum $X$ is of the form

$X \to \cdots \to L_n X \to L_{n-1} X \to \cdots \to L_0 X \,.$

The homotopy fibers of each stage of the tower

$M_n(X) \coloneqq fib(L_{E(n)}X \longrightarrow L_{E(n-1)}(X))$

is called the $n$th monochromatic layer of $X$.

The chromatic convergence theorem states mild conditions under which the homotopy limit over this tower is the $p$-localization

$X \to X_{(p)}$

of $X$.

Since moreover $L_n X$ is the homotopy fiber product

$L_n X \simeq L_{K(n)}X \underset{L_{n-1}L_{K(n)}X}{\times} L_{n-1}X$

(see at smash product theorem) it follows that in principle one can study a spectrum $X$ by understanding all its “chromatic pieces” $L_{K(n)} X$. This is the topic of chromatic homotopy theory.

## Properties

### Chromatic spectral sequence

The spectral sequence of a filtered stable homotopy type associated with the chromatic tower (regarded as a filtered object in an (infinity,1)-category) is the chromatic spectral sequence (Wilson 13, section 2.1.2)

## Examples

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 spectrum
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$BPR-theory
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$complex cobordism cohomologyMUMR-theory

## References

Last revised on November 18, 2013 at 21:18:42. See the history of this page for a list of all contributions to it.