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.

For years I have been echoing my betters, especially Mike Hopkins, and telling anyone who would listen that the chromatic picture of stable homotopy theory is dictated and controlled by the geometry of the moduli stack Mfg of smooth, one-dimensional formal groups. Specifically, I would say that the height filtration of Mfg dictates a canonical and natural decomposition of a quasi-coherent sheaf on Mfg, and this decomposition predicts and controls the chromatic decomposition of a finite spectrum. This sounds well, and is even true, but there is no single place in the literature where I could send anyone in order for him or her to get a clear, detailed, unified, and linear rendition of this story. This document is an attempt to set that right. Before going on to state in detail what I actually hope to accomplish here, I should quickly acknowledge that the opening sentences of this introduction and, indeed, this whole point of view is not original with me. I have already mentioned Mike Hopkins, and just about everything I’m going to say here is encapsulated in the table in section 2 of [15] and can be gleaned from the notes

this article we will generalise both results to an arbitrary homotopy category associated to a Waldhausen category. Let us begin by reminding the reader about the formalism of a Waldhausen category. Note that our presentation here will be very sketchy and incomplete. Throughout this article we will assume familiarity with Waldhausen 's foundational article [11]. A Waldhausen category will mean a category with cofibrations and weak equivalences, satisfying the Gluing Lemma, Extension Axiom, Saturation Axiom and Cylinder Axiom, as in Waldhausen's article [11]. Starting with such a category, one can construct an associated stable homotopy category, by inverting the suspension functor and the weak equivalences. Inverting the suspension functor is harmless; the resulting category is still Waldhausen. For a discussion, see Section 1. Inverting the weak equivalences is drastic. One obtains something which is decidedly not a Waldhausen category, but only a triangulated category. Throughout the article we will assume that the Waldhausen categories we are dealing with have an invertible suspension functor. As we have already said, this is harmless once we replace a Waldhausen category by its stabilisation. For the uninitiated, let us briefly recall what a Waldhausen category is. It is a category C, with two subcategories w(C) and c(C). The objects in the three categories are the same. The morphisms in w(C) are called weak equivalences and denoted by the letter w, the morphisms in c(C) are called cofibrations and denoted by the letter c. 2 These must satisfy a long list of axioms, which we do not want to repeat here. Perhaps the most important is that pushouts of cofibrations exist. That is, given a diagram

Abstract. This very speculative talk suggests that a theory of fundamental groupoids for tensor triangulated categories could be used to describe the ring of integers as the singular fiber in a family of ring-spectra parametrized by a structure space for the stable homotopy category, and that Bousfield localization might be part of a theory of ‘nearby ’ cycles for stacks or orbifolds. One of the motivations for this talk comes from John Rognes ’ Galois theory for structured ring spectra. His paper [37] ends with some very interesting remarks about analogies between classical primes in algebraic number fields and the non-Euclidean primes of the stable homotopy category, and I try here to develop a

The two categories are not Quillen equivalent, and his proof uses systems of triangulated diagram categories rather than model categories. Our main result is that in the case n = 1 Franke’s functor maps the derived tensor product to the smash product. It can however not be an associative equivalence of monoidal categories. The first part of our paper sets up a monoidal version of Franke’s systems of triangulated diagram categories and explores its properties. The second part applies these results to the specific construction of Franke’s functor in order to prove the above result. 1.

ABSTRACT. The additivity theorem for dérivateurs associated to complicial biWaldhausen categories is proved. Also, to any exact category in the sense of Quillen a K-theory space is associated. This K-theory is shown to satisfy the additivity, approximation and resolution theorems. 1.

He spent 11 years in a succession of postdoctoral positions in Canada, USA and Britain, including two years at the University of Chicago as an L. E. Dickson Instructor and 2 years as an EPSRC Advanced Fellow, before being appointed in 1991 to a Lectureship (and subsequently in 1996 to a Readership) at the University of Glasgow. His research has centred on algebraic topology, especially stable homotopy theory. In particular he has focused on applications of algebra and number theory to complex oriented and periodic cohomology theories (especially K-theory and elliptic cohomology). For a representative overview of his work see [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. In recent years he has been very involved in work on structured ring spectra and related topics, and organised a series of workshops in Glasgow, Bonn and Rosendal (Norway), as well as editing a book based on the first of these [13]. Sarah Whitehouse was awarded a Ph.D. from the University of Warwick in 1994. She spent several years in France, as a Marie-Curie post-doctoral researcher at the Université Paris-Nord and as a Lecturer at the Université d’Artois. She joined the University of Sheffield as a Lecturer in 2002 and was promoted to Senior Lecturer in 2005. Much of her work has involved the algebras of operations or cooperations of generalised cohomology theories [18, 19, 27, 38]. Recently, this has given new results for complex K-theory, cobordism and the Morava Ktheories

complexes with entries in E. Shifting a complex by 3 positions yields an outer shift

An overview of abelian varieties in homotopy theory

The stable homotopy category has been extensively studied by algebraic topologists for a long time. For many applications it is convenient or even necessary to work with point set level models of spectra as opposed to working up-to-homotopy, and the outcome of a calculation might depend