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19
Morava Ktheories and localisation
 MEM. AMER. MATH. SOC
, 1999
"... 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 ..."
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Cited by 101 (19 self)
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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.
Equivariant Formal Group Laws.
 Proc. London Math. Soc 81
"... . Motivated by complex oriented theories we define Aequivariant formal group laws for any abelian compact Lie group A, show there is a representing ring for them, and begin the investigation of it. We examine a number of topological cases, including Ktheory in some detail. JPCG thanks Hal Sadofsky ..."
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. Motivated by complex oriented theories we define Aequivariant formal group laws for any abelian compact Lie group A, show there is a representing ring for them, and begin the investigation of it. We examine a number of topological cases, including Ktheory in some detail. JPCG thanks Hal Sadofsky for many useful conversations, the University of Chicago and the University of Oregon for their hospitality during May 1996, when this work was begun, and the Nuffield Foundation for its support. 1. Introduction The purpose of this article is to formulate and study the notion of equivariant formal group law appropriate for understanding orientable complex stable equivariant cohomology theories E A (\Delta), at least when the compact Lie group A of equivariance is abelian. The aim is to understand cohomology theories which behave in a simple way on complex vector bundles, and hence give rise to a good theory of characteristic classes. In particular we want to understand tom Dieck's homot...
Formal schemes and formal groups
 in honor of J.M. Boardman, volume 239 of Contemporary Mathematics
, 1999
"... 1.1. Notation and conventions 3 1.2. Even periodic ring spectra 3 2. Schemes 3 ..."
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Cited by 12 (6 self)
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1.1. Notation and conventions 3 1.2. Even periodic ring spectra 3 2. Schemes 3
Morava Etheory of symmetric groups
 Topology
, 1998
"... Abstract. We compute the completed E(n) cohomology of the classifying spaces of the symmetric groups, and relate the answer to the theory of finite subgroups of formal groups. 1. ..."
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Abstract. We compute the completed E(n) cohomology of the classifying spaces of the symmetric groups, and relate the answer to the theory of finite subgroups of formal groups. 1.
Tate cohomology of theories with one dimensional coecient ring
, 1995
"... Abstract. For a finite group G we calculate the ETate cohomology t(E)∗G and the Ehomology E∗(BG+) as functors of the augmented commutative ring E∗(BG+) when E∗(·) is a complex oriented, vn periodic cohomology theory with one dimensional graded coefficient ring E∗. 1. Introduction. Let G be a finit ..."
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Abstract. For a finite group G we calculate the ETate cohomology t(E)∗G and the Ehomology E∗(BG+) as functors of the augmented commutative ring E∗(BG+) when E∗(·) is a complex oriented, vn periodic cohomology theory with one dimensional graded coefficient ring E∗. 1. Introduction. Let G be a finite group and p be a prime. Our aim is to calculate the coefficient ring of the Gequivariant Tate theory for a complex oriented vnperiodic theory E. In [7] we showed that if E is mod p Morava Ktheory (whose coefficient ring K(n) ∗ = Fp[vn, v−1n] is a graded field) then the associated Tate theory is trivial and the representing spectrum is
Equivariant Formal Group Laws and Complex Oriented Cohomology Theories
"... The article gives an introduction to equivariant formal group laws, and explains its relevance to complex oriented cohomology theories in general and to complex cobordism in particular. Contents 1. ..."
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The article gives an introduction to equivariant formal group laws, and explains its relevance to complex oriented cohomology theories in general and to complex cobordism in particular. Contents 1.
Local Cohomology In Equivariant Topology
"... The article describes the role of local homology and cohomology in understanding the equivariant cohomology and homology of universal spaces. This brings to light an interesting duality property related to the Gorenstein condition. The phenomena are studied and illustrated in several rather di ffren ..."
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The article describes the role of local homology and cohomology in understanding the equivariant cohomology and homology of universal spaces. This brings to light an interesting duality property related to the Gorenstein condition. The phenomena are studied and illustrated in several rather di ffrent families of examples. Both topology and commutative algebra benefit from the connection, and many interesting questions remain open.
HKR CHARACTERS AND HIGHER TWISTED SECTORS
, 2002
"... Abstract. This is an expository talk, presented at the ChengDu (Sichuan) ICM Satellite conference on stringy orbifolds. It is intended as an introduction to the work of Hopkins, Kuhn, and Ravenel on generalized group characters, which seems to fit very well with the theory of what physicists call hi ..."
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Abstract. This is an expository talk, presented at the ChengDu (Sichuan) ICM Satellite conference on stringy orbifolds. It is intended as an introduction to the work of Hopkins, Kuhn, and Ravenel on generalized group characters, which seems to fit very well with the theory of what physicists call higher twisted sectors in the theory of orbifolds. I would like to acknowledge many conversations with Matthew Ando about the contents of this paper. In a better world, he would be its coauthor.
THE BP 〈N 〉 COHOMOLOGY OF ELEMENTARY ABELIAN GROUPS
, 2000
"... In this paper we define three elements of a certain generalised cohomology ring BP 〈m, n 〉 ∗ BVk. Here m, n and k are nonnegative integers with k + m ≤ n + 1, there is a fixed prime p not exhibited in the notation, and Vk is an elementary Abelian pgroup of rank k. We show that these elements are e ..."
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In this paper we define three elements of a certain generalised cohomology ring BP 〈m, n 〉 ∗ BVk. Here m, n and k are nonnegative integers with k + m ≤ n + 1, there is a fixed prime p not exhibited in the notation, and Vk is an elementary Abelian pgroup of rank k. We show that these elements are equal; this is striking, because the three definitions are very different. The significance of our equation is not yet entirely clear, but