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
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).
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