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31
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 38 (9 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
On the topological Hochschild homology of bu. I.
 AMER. J. MATH
, 1993
"... 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 plocal connective Ktheory (see for example [25]) and THH is the topological Hochschild homology ..."
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Cited by 34 (0 self)
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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 plocal connective Ktheory (see for example [25]) and THH is the topological Hochschild homology
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 27 (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
The stack of formal groups in stable homotopy theory
 Adv. Math
"... We construct the algebraic stack of formal groups and use it to provide a new perspective onto a recent result of M. Hovey and N. Strickland on comodule categories for Landweber exact algebras. This leads to a geometric understanding of their results as well as to a generalisation. 1. ..."
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Cited by 25 (4 self)
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We construct the algebraic stack of formal groups and use it to provide a new perspective onto a recent result of M. Hovey and N. Strickland on comodule categories for Landweber exact algebras. This leads to a geometric understanding of their results as well as to a generalisation. 1.
Comodules and Landweber exact homology theories
 Adv. Math
"... Abstract. We show that, if E is a commutative MUalgebra spectrum such ..."
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Cited by 23 (1 self)
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Abstract. We show that, if E is a commutative MUalgebra spectrum such
Morita theory for Hopf algebroids and presheaves of groupoids
 Amer. J. Math
"... Abstract. Comodules over Hopf algebroids are of central importance in algebraic topology. It is wellknown that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff, the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasicoheren ..."
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Cited by 17 (2 self)
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Abstract. Comodules over Hopf algebroids are of central importance in algebraic topology. It is wellknown that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff, the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasicoherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology T on Aff give rise to equivalences of categories of sheaves in that topology. We then show using faithfully flat descent that an internal equivalence in the flat topology gives rise to an equivalence of categories of quasicoherent sheaves. The corresponding statement for Hopf algebroids is that weakly equivalent Hopf algebroids have equivalent categories of comodules. We apply this to formal group laws, where we get considerable generalizations of the MillerRavenel [MR77] and HoveySadofsky [HS99] change of rings theorems in algebraic topology.
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 17 (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.
The Bous spectral sequence for periodic homology theories
"... Abstract. We construct the BouseldKan (unstable Adams) spectral sequence based on certain nonconnective periodic homology theories E such as complex periodic Ktheory, and dene an Ecompletion of a space X. For X = S2n+1 and E = K we calculate the E2term and show that the spectral sequence conv ..."
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Cited by 13 (3 self)
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Abstract. We construct the BouseldKan (unstable Adams) spectral sequence based on certain nonconnective periodic homology theories E such as complex periodic Ktheory, and dene an Ecompletion of a space X. For X = S2n+1 and E = K we calculate the E2term and show that the spectral sequence converges to the homotopy groups of the Kcompletion of the sphere. This also determines all of the homotopy groups of the (unstable) Ktheory localization of S2n+1 including three divisible groups in negative stems. 1.
Morava modules and BrownComenetz duality
 Amer. J. Math
, 1997
"... Abstract. Fix a prime number p, and let X be a finite spectrum whose (n 1)st Morava Ktheory is trivial but whose nth Morava Ktheory is nontrivial, n> 0. We prove, following a method outlined to us by M. J. Hopkins, that, if 2p> n2 + n + 2, the Morava module of the BrownComenetz dual of the ..."
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Cited by 12 (0 self)
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Abstract. Fix a prime number p, and let X be a finite spectrum whose (n 1)st Morava Ktheory is trivial but whose nth Morava Ktheory is nontrivial, n> 0. We prove, following a method outlined to us by M. J. Hopkins, that, if 2p> n2 + n + 2, the Morava module of the BrownComenetz dual of the E(n)localization of X is isomorphic to a suspension of the Pontryagin dual of the Morava module of X. To complete this proof, we found it necessary to develop a more canonical construction of certain modified Adams spectral sequences; this construction should be of independent interest. Introduction. Let X be a K(n 1)acyclic finite spectrum, n 1, and let LnX denote its E(n)localization. As usual, K(n 1) denotes the (n 1)st
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 st ..."
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Cited by 10 (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...