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.

1.2. The axioms

Abstract. In [AHS01] the authors constructed a natural map, called the sigma orientation, from the Thom spectrum MU〈6 〉 to any elliptic spectrum in the sense of [Hop95]. MU〈6 〉 is an H ∞ ring spectrum, and in this paper we show that if (E, C, t) is the elliptic spectrum associated to the universal deformation of a supersingular elliptic curve over a perfect field of characteristic p> 0, then the sigma orientation is a map of H ∞ ring spectra.

Given a good homology theory E and a topological space X, E∗X is not just an E∗-module but also a comodule over the Hopf algebroid (E∗, E∗E). We establish a framework for studying the homological algebra of comodules over a well-behaved Hopf algebroid (A, Γ). That is, we construct

this paper. The Conner-Floyd theorem was later generalized by Landweber in his exact functor theorem [Lan]. Conner and Floyd also prove symplectic bordism determines real K-theory: MSp

. This is the first of two interconnected parts: Part I contains the geometric theory of generalized modular forms and their connections with the cooperation algebra for elliptic cohomology, E" E", while Part II is devoted to the more algebraic theory associated with Hecke algebras and stable operations in elliptic cohomology. We investigate the structure of the stable operation algebra E" E" by first determining the dual cooperation algebra E" E". A major ingredient is our identification of the cooperation algebra E" E" with a ring of generalized modular forms whoses exact determination involves understanding certain integrality conditions; this is closely related to a calculation by N. Katz of the ring of all `divided congruences' amongst modular forms. We relate our present work to previous constructions of Hecke operators in elliptic cohomology. We also show that a well known operator on modular forms used by Ramanujan, Swinnerton-Dyer, Serre and Katz cannot extend to a stabl...

. We consider a generalization of the Johnson--Wilson spectrum E(n) for which E(n) is a local ring with maximal ideal In . In particular we prove that the spectra E(n), E(n) and [ E(n) are Bousfield equivalent. We also show that the Hopf algebroid E(n) E(n) is a free E(n) -module, generalizing a result of Adams and Clarke for KU KU . Introduction For each prime p and n ? 0, the Johnson--Wilson ring spectrum E(n) provides an important example of a p-local periodic ring spectrum. The associated Hopf algebroid E(n) E(n) is well known to be flat over E(n) , but as far as we can determine there is no proof in the literature that it is a free module for every n. Of course, after passage to the I n -adic completion [ E(n), and more drastically the I n -adic completion of E(n) E(n) (see [4, 8]), such problems disappear. On the other hand, for the ring spectrum KU , the associated Hopf algebroid KU KU was shown to be free over KU by Frank Adams and Francis Clarke [3, 2, 6]. Actuall...