Chromatic convergence and its discontents

The chromatic convergence theorem of Ravenel and Hopkins asserts that, if $X$ is a $p$-local finite spectrum, then the homotopy limit $\text{holim}_n L_{E(n)}(X)$ of the localizations of $X$ at each of the Johnson-Wilson $E$-theories $E(n)$ is homotopy-equivalent to $X$ itself. One way of seeing the chromatic convergence theorem is by regarding the functor sending a spectrum $X$ to $\text{holim}_n L_{E(n)}(X)$ as a kind of completion, "chromatic completion," which has the agreeable property that $p$-local finite spectra are all already chromatic complete. Then there are two natural questions:

1. Given a (not necessarily finite) spectrum $X$, is there a criterion that lets us decide easily whether $X$ is chromatically complete or not?

2. Given a nonclassical setting for homotopy theory, such as equivariant spectra or motivic spectra, what analogue of the chromatic convergence theorem might hold?

We give an answers to each of these two questions. For a symmetric monoidal stable model category $C$ satisfying some reasonable hypotheses, we produce a natural notion of "chromatic completion," as well as the notion of a "chromatic cover," a commutative monoid object which shares important properties with the complex cobordism spectrum $MU$ from classical stable homotopy theory. We show that, if a chromatic cover exists in $C$, then any object $X$ satisfying Serre’s condition $S_n$ for any $n$ is chromatically complete if and only if the microlocal cohomology of $X$ vanishes. (Of course we have to define Serre’s condition $S_n$ as well as microlocal cohomology in this context!)

We get two important corollaries: first, by computing some microlocal cohomology groups, we find that large classes of non-finite classical spectra are not chromatically complete, such as the connective spectra $ku$ and $BP\langle n \rangle$ for all finite $n$. We also get some non-chromatic-completeness results for $\text{ko}$, $\text{tmf}$, and $\text{taf}$. Second, we get conditions under which a chromatic completion theorem can hold for motivic and equivariant spectra: one needs a chromatic cover to exist in those categories of spectra. We identify a candidate for such a chromatic cover for motivic spectra over $\text{Spec}\, C$, assuming the Dugger-Isaksen nilpotence conjecture.