nLab
chromatic convergence theorem

Contents

Context

Stable Homotopy theory

Higher algebra

Contents

Idea

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

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

where K(n)K(n) is the nnth Morava K-theory (for the given prime pp). These arrange into the chromatic tower which for each spectrum XX is of the form

XL nXL n1XL 0X. 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 pp-localization

XX (p) X \to X_{(p)}

of XX.

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

Since moreover L nXL_n X is the homotopy fiber product

L nXL K(n)X×L n1L K(n)XL n1X 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 XX by understanding all its “chromatic pieces” L K(n)XL_{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.