nLab chromatic convergence theorem

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.

Related concepts

• chromatic layer

• smash product theorem

References

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 32 The chromatic convergence theorem (pdf)