# nLab chromatic convergence theorem

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

$L_n \coloneqq L_{K(0)\vee \cdots \vee K(n)} \,,$

where $K(n)$ is the $n$th Morava K-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 chromatic convergence theorem states mild conditions under which the homotopy limit over this tower is the $p$-localization

$X \to X_{(p)}$

of $X$.

In particular if $X$ is a finite p-local spectrum then the chromatic convergence theorem says that the homotopy limit over the chromatic tower of $X$ reproduces $X$.

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 and see this remark at fracture square ) it follows that in principle one may study any spectrum $X$ by understanding all its “chromatic pieces” $L_{K(n)} X$. This is the topic of chromatic homotopy theory.

## References

Last revised on December 14, 2015 at 09:23:19. See the history of this page for a list of all contributions to it.