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35
Morava Ktheories and localisation
 Mem. Amer. Math. Soc
, 1999
"... 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)nilpoten ..."
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Cited by 63 (18 self)
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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 BrownComenetz 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.
Bousfield localization functors and Hopkins’ chromatic splitting conjecture
 In Proceedings of the Čech centennial homotopy theory conference
, 1993
"... This paper arose from attempting to understand Bousfield localization functors in stable homotopy theory. All spectra will be plocal for a prime p throughout this paper. Recall that if E is a spectrum, a spectrum X is Eacyclic if E ∧ X is null. A spectrum is Elocal if every map from an Eacyclic ..."
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Cited by 33 (9 self)
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This paper arose from attempting to understand Bousfield localization functors in stable homotopy theory. All spectra will be plocal for a prime p throughout this paper. Recall that if E is a spectrum, a spectrum X is Eacyclic if E ∧ X is null. A spectrum is Elocal if every map from an Eacyclic spectrum
Invertible spectra in the E(n)local stable homotopy category
 J. London Math. Soc
"... 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 inver ..."
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Cited by 28 (7 self)
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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
Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence
, 139
"... 1.2. The axioms ..."
Products on MUmodules
 Trans. Amer. Math. Soc
, 1999
"... 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 ..."
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Cited by 23 (5 self)
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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 BaasSullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising 2local MU∗modules as MUmodules. 1.
The sigma orientation is an H∞ map
 American Journal of Mathematics
"... 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 d ..."
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Cited by 15 (2 self)
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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.
Homotopy theory of comodules over a Hopf algebroid
, 2003
"... 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 wellbehaved Hopf algebroid (A, Γ). That is, we construct ..."
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Cited by 13 (3 self)
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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 wellbehaved Hopf algebroid (A, Γ). That is, we construct
Spin Cobordism Determines Real KTheory
 Math. Z
, 1995
"... this paper. The ConnerFloyd theorem was later generalized by Landweber in his exact functor theorem [Lan]. Conner and Floyd also prove symplectic bordism determines real Ktheory: MSp ..."
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Cited by 10 (3 self)
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this paper. The ConnerFloyd theorem was later generalized by Landweber in his exact functor theorem [Lan]. Conner and Floyd also prove symplectic bordism determines real Ktheory: MSp
Operations and Cooperations in Elliptic Cohomology, Part I: Generalized modular forms and the cooperation algebra
, 1995
"... . 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 opera ..."
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Cited by 9 (7 self)
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. 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, SwinnertonDyer, Serre and Katz cannot extend to a stabl...
InLocal JohnsonWilson Spectra and their Hopf algebroids
 DOCUMENTA MATH.
, 2000
"... We consider a generalization E(n) of the JohnsonWilson spectrum E(n) for which E(n) ∗ is a local ring with maximal ideal In. We prove that the spectra E(n), E(n) and Ê(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 ..."
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Cited by 8 (4 self)
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We consider a generalization E(n) of the JohnsonWilson spectrum E(n) for which E(n) ∗ is a local ring with maximal ideal In. We prove that the spectra E(n), E(n) and Ê(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.