In this paper, I write up a lot of facts about Bousfield classes known
previously to Mike Hopkins, and I discover some new ones. Stating the
chromatic splitting conjecture precisely is probably the most important
feature of the paper, but I was pretty happy with some of the results as
well. In particular, I show that a finite torsion spectrum is local
with respect to any infinite collection of Morava K-theories, and that
the only smashing localization that doesn't kill any finites is the
identity (p-locally). This paper is probably the clearest statement so
far of Mike's philosophy that K(n)-localization is the fundamental
category for stable homotopy theorists to understand. However, it must
be confessed that this philosophy has not been fully borne out as yet.
In particular, I expected people to make at least a little progress on
the chromatic splitting conjecture, and that has not happened. Ethan
Devinatz has shown its relevance to Freyd's generating hypothesis, which
is one of the main reasons Mike made the conjecture.
By the way, there is a mistake in this paper. In the appendix I
claim inverse limits of cofiber sequences are cofiber sequences, but
that is wrong in general. In the particular case I need it for, it
still works, but I won't prove it for you unless you really want me to.