# nLab monochromatic layer

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 $p$). 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 $X$.

## Examples

### Chromatic layers of the sphere spectrum

Discussion of the first chromatic layer of the sphere spectrum is due to (Miller-Ravenel-Wilson 77). Review is in (Knudsen 13). This is the image of the J-homomorphism.

## References

General lecture notes include

Discussion of the chromatic layers of the sphere spectrum is in

Lecture notes on this include

• Ben Knudsen, First chromatic layer of the sphere spectrum = homotopy of the $K(1)$-local sphere, talk at 2013 Pre-Talbot Seminar (pdf)

Last revised on November 18, 2013 at 06:30:21. See the history of this page for a list of all contributions to it.