Ravenel conjectures

Statement

Fix a prime number $p$. Let $K(n)$ be the $n$-th Morava K-theory, with $K(0) = H\Q$. Let $L_n$ be Bousfield localization with respect to the Johnson–Wilson spectrum $E(n)$, as in chromatic homotopy theory. A finite $p$-local spectrum $X$ has type $n$ if $K(n)_* X \ne 0$ and $K(m)_* X = 0$ for all $m < n$. The following conjectures of Ravenel are theorems.

1. (Nilpotence) A self-map $f \colon \Sigma^d X \to X$ of a finite $p$-local spectrum is nilpotent if $K(n)_*(f)$ is nilpotent for every $n \ge 0$. This is a form of the nilpotence theorem.
2. (Periodicity) If $X$ has type $n \ge 1$, there is a map $v \colon \Sigma^d X \to X$ such that $K(n)_*(v)$ is an isomorphism and $K(m)_*(v)$ is nilpotent for $m \ne n$.
3. (Thick subcategories) Let $\mathcal{C}_n$ be the category of finite $p$-local spectra of type at least $n$. Every thick subcategory of $\mathcal{C}_0$ is either some $\mathcal{C}_n$ or the zero subcategory.
4. (Smash product) For every spectrum $X$, the natural map $X \wedge L_n S \to L_n X$ is an equivalence, where $S$ is the $p$-local sphere spectrum.
5. (Chromatic convergence) For every finite $p$-local spectrum $X$, the map $X \to \operatorname{holim}_n L_n X$ is an equivalence.