Abstract. We study the structure of the categories of K(n)-local and E(n)local spectra, using the axiomatic framework developed in earlier work of the authors with John Palmieri. We classify localising and colocalising subcategories, and give characterisations of small, dualisable, and K(n)-nilpotent spectra. We give a number of useful extensions to the theory of vn self maps of finite spectra, and to the theory of Landweber exactness. We show that certain rings of cohomology operations are left Noetherian, and deduce some powerful finiteness results. We study the Picard group of invertible K(n)-local spectra, and the problem of grading homotopy groups over it. We prove (as announced by Hopkins and Gross) that the Brown-Comenetz dual of MnS lies in the Picard group. We give a detailed analysis of some examples when n =1 or 2, and a list of open problems.

This paper arose from attempting to understand Bousfield localization functors in stable homotopy theory. All spectra will be p-local for a prime p throughout this paper. Recall that if E is a spectrum, a spectrum X is E-acyclic if E ∧ X is null. A spectrum is E-local if every map from an E-acyclic spectrum

Suppose C is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in C, and the Picard group Pic(C) is the collection of isomorphism classes of such invertible objects. The

modules over highly structured ring spectra to give new constructions of MUmodules such as BP, K(n) and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over MU [ 1] ∗ that are concentrated in 2 degrees divisible by 4; this guarantees that various obstruction groups are trivial. We extend these results to the cases where 2 = 0 or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising 2-local MU∗-modules as MU-modules. 1.

The work to be presented in this paper has been inspired by several of Professor Graeme Segal’s papers. Our search for a geometrically defined elliptic cohomology theory with associated elliptic objects obviously stems from his Bourbaki seminar [Se88]. Our readiness to form group completions of symmetric monoidal categories

We construct stable operations Tn : E" ( ) \Gamma! E"(1=n) ( ) for n ? 0 in the version of elliptic cohomology where the coefficient ring E" agrees with the ring of modular forms for SL 2 (Z) which are meromorphic at 1, and Tn restricts to the n th Hecke operator Tn on E" . In the past few years, the idea of elliptic cohomology has emerged from the combined efforts of a variety of mathematicians and physicists, and it is widely expected that it will play as important a role in global analysis and topology as K--theory and bordism have in the past. At present, there is no explicit geometric description of the cohomology theories that arise in this area, although there are several promising ideas which it is hoped will eventually lead to such a description. On the other hand, there are constructions of these theories based upon cobordism theories and for many purposes these seem to be adequate, at least for problems within the realm of stable homotopy theory. In particular, in ...

This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which non-specialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 4

Let ℓp be the p-complete connective Adams summand of topological K-theory, with coefficient ring (ℓp) ∗ = Zp[v1], and let V (1) be the Smith–Toda complex, with BP∗(V (1)) = BP∗/(p, v1). For p ≥ 5 we explicitly compute the V (1)-homotopy of the algebraic K-theory spectrum of ℓp, denoted V (1)∗K(ℓp). In particular we find that it is a free finitely generated module over the polynomial algebra P (v2), except for a sporadic class in degree 2p − 3. Thus also in this case algebraic K-theory increases chromatic complexity by one. The proof uses the cyclotomic trace map from algebraic K-theory to topological cyclic homology, and the calculation is

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The purpose of this paper and its sequel is to determine the homotopy groups of the spectrum THH(l). Here p is an odd prime, l is the Adams summand of p-local connective K-theory (see for example [25]) and THH is the topological Hochschild homology