symmetric monoidal (∞,1)-category of spectra
For each prime $p \in \mathbb{N}$ and for each natural number $n \in \mathbb{N}$ there is a Bousfield localization of spectra
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
The chromatic convergence theorem states mild conditions under which the homotopy limit over this tower is the $p$-localization
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
(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.
Last revised on December 14, 2015 at 14:23:19. See the history of this page for a list of all contributions to it.