2

M. HOVEY AND N. P. STRICKLAND

In the second half of the paper, we concentrate on 3C. In Section 7, we prove our

most basic results about the .ftf(n)-local category X. In particular, we prove that

it is irreducible, in the sense that the only localising or colocalising subcategories

are {0} and X itself. We also study the localisation functor L = LK and some

related functors, describing them in terms of towers of generalised Moore spectra.

In Section 8 we study two different notions of finiteness in 3C, called smallness and

dualisability* The (local) sphere is dualisable but not small; some rather unexpected

spectra are dualisable, such as the localisation of BG for a finite group G. We prove

a number of different characterisations of smallness and dualisability; in particular,

we give convenient tests in terms of computable cohomology theories. We also show

that dualisable spectra lie in the thick subcategory generated by

E1,

and that X-

small spectra lie in the thick subcategory generated by K. In Section 9 we study

homology and cohomology theories on 3C, and prove that both are representable

in a suitable sense. In Section 10 we study a version of Brown-Comenetz duality

appropriate to the iif(n)-local setting, and we prove that the Brown-Comenetz dual

of the monochromatic sphere is an element of the Picard group. This result was

stated in [HG94], In Section 11, we introduce a natural topology on the groups

[X, Y] for X and Y in 3C, and prove a number of properties. In Section 12, we

return to the study of the category D of dualisable spectra. We prove a nilpotence

theorem and an analogue of the Krull-Schmidt theorem, saying that every dualisable

spectrum can be written as a wedge of indecomposables in an essentially unique

way. We make some remarks about ideals in 2), but we have not been able to prove

the obvious conjectures about them. In Section 13 we study if-nilpotent spectra,

proving a number of interesting characterisations of them. In Section 14 we study

the problem of grading homotopy groups over the Picard group of invertible spectra,

rather than just over the integers. We have a satisfactory theory for homotopy

groups of X-small spectra, but the general case seems less pleasant. We show that

the Picard group is profinite in a precise sense, but we do not know if it is finitely

generated over the p-adics. Section 15 is devoted to the study of the simplest

examples. Even when n = 1 and p is odd, there are simple counterexamples to

plausible conjectures. We also consider the case n = 2 and p 3, showing that

the Picard graded homotopy groups of the Moore spectrum are mostly infinite.

We conclude the main body of the paper with Section 16, which contains a list of

interesting questions that we have been unable to answer, some of them old and

others new.

We also have two appendices: the first addresses the sense in which the E-

cohomology of a if-local spectrum is complete, as mentioned above, and the second

shows that some other interesting localisations of the category of spectra have a

rather different behaviour, in that they contain no nonzero small objects at all.

We have chosen to rely on a minimum of algebraic prerequisites; in particular,

we say almost nothing about formal groups or the Morava stabiliser groups. Sec-

tion 2.3 is thus considerably less intuitive than it should be, but we did not wish to

double the length of the paper by a careful exposition of the relationship between

Morava U-theory and the Morava stabilizer groups. We have preferred to use thick

subcategory arguments rather than spectral sequences where possible. We have

chosen our methods very carefully to avoid having to say anything special when

p = 2. For this reason we have generally used E rather than if, as E is commutative

at all primes (for example).