INTRODUCTION

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

subcategories.

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.

1