The stable homotopy category 8 is extraordinarily complicated. However, there
is a set of approximations to it that are much simpler and closer to algebra. The
stable homotopy category is somewhat analogous to the derived category of a ring i?,
except that R is replaced by the stable sphere 5°. In practice, we always consider
the p-local stable homotopy category and the p-local sphere for some prime p,
without changing the notation. It is very common to study i?-modules using the
fields over i?, and in recent years there has been much work on studying the stable
homotopy category via its fields. These fields are referred to as Morava if-theories,
denoted by K(n), and were introduced by Morava in the early 1970's. See [Mor85]
for a description of Morava's earlier work.
Associated to the Morava if-theories are various (homotopy) categories of local
spectra that are the approximations to the stable homotopy category mentioned
above. There is the category £ of spectra local with respect to K(0) V V K(n)
and the category % of spectra local with respect to K{n). (We will always have a
fixed n with 0 n oo in mind in this paper). These categories are themselves
stable homotopy categories, in the sense of [HPS97]. The purpose of this paper is
to study the structure of these categories. We will show that the category % is in
a certain sense irreducible; it has no nontrivial further localisations. On the other
hand, there are a number of results (such as the Chromatic Convergence Theorem of
Hopkins and Ravenel [Rav92a], or the proof by the same authors that suspension
spectra are harmonic [HR92]) which indicate that an understanding of % for all
n and p will give complete information about 8. Hopkins' Chromatic Splitting
Conjecture [Hov95a], if true, would be a still stronger result in this direction.
The first half of the paper is mostly concerned with issues we need to resolve
before beginning our study of X. In Section 1, we define the basic objects of
study: the ring spectra E(n), E = E(n) and K = K(n) the categories £ and
% and so on. Here and throughout the paper we use the results of [EKMM96]
because this method is both more elegant and more powerful than the usual Bass-
Sullivan construction. In Section 2, we study .E-cohomology. We prove that E*X
is complete, using a slightly modified notion of completeness which turns out to be
more appropriate than the traditional one. We show that a well-known quotient of
the ring of operations in .E-cohomology is a non-commutative Noetherian local ring.
We also show that there are no even degree phantom maps between evenly graded
Landweber exact spectra such as JB, a result that has been speculated about for a
long time. In Section 3, we prove some basic results about the (very simple) category
of X-injective spectra. We then turn in Section 4 to the study of generalised Moore
spectra, and prove a number of convenient and enlightening extensions to the theory
developed by Hopkins and Smith [HS, Rav92a] and Devinatz [Dev92]. In Section 5,
we assemble some (mostly well-known) results about the Bousfield classes of a
number of spectra that we will need to study. In Section 6, we study the i?(n)-local
category £. We prove some results about nilpotence, and we classify the thick
subcategories of small objects, the localising subcategories and the colocalising
The first author was partially supported by an NSF Postdoctoral Fellowship.
The second author was partially supported by an NSF Grant.
Received by the editor May 15, 1997.
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