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18
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
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 14 (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.
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
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 12 (1 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 12 (1 self)
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Abstract. We show that, if E is a commutative MUalgebra spectrum such
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.
Isogenies Of Supersingular Elliptic Curves Over Finite Fields And Operations In Elliptic Cohomology
"... . In this paper we investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields. Our main results provide a framework in which we give a conceptually simple new proof of an elliptic cohomology version of the Morava change of r ..."
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Cited by 6 (3 self)
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. In this paper we investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields. Our main results provide a framework in which we give a conceptually simple new proof of an elliptic cohomology version of the Morava change of rings theorem and also gives models for explicit stable operations in terms of isogenies and morphisms in certain enlarged isogeny categories. We are particularly inspired by number theoretic work of G. Robert, whose work we reformulate and generalize in our setting. Introduction In previous work we investigated supersingular reductions of elliptic cohomology [5], stable operations and cooperations in elliptic cohomology [3, 4, 6, 8] and in [9, 10] gave some applications to the Adams spectral sequence based on elliptic (co)homology. In this paper we investigate stable operations in supersingular elliptic cohomology using isogenies of supersingular elliptic curves over finite fields; this is ...
On degeneration of onedimensional formal group laws and stable homotopy theory, AJM 125
, 2003
"... Abstract. In this note we study a certain formal group law over a complete discrete valuation ring F[[un−1]] of characteristic p> 0 which is of height n over the closed point and of height n − 1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on t ..."
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Cited by 5 (0 self)
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Abstract. In this note we study a certain formal group law over a complete discrete valuation ring F[[un−1]] of characteristic p> 0 which is of height n over the closed point and of height n − 1 over the generic point. By adjoining all coefficients of an isomorphism between the formal group law on the generic point and the Honda group law Hn−1 of height n − 1, we get a Galois extension of the quotient field of the discrete valuation ring with Galois group isomorphic to the automorphism group Sn−1 of Hn−1. We show that the automorphism group Sn of the formal group over the closed point acts on the quotient field, lifting to an action on the Galois extension which commutes with the action of Galois group. We use this to construct a ring homomorphism from the cohomology of Sn−1 to the cohomology of Sn with coefficients in the quotient field. Applications of these results in stable homotopy theory and relation to the chromatic splitting conjecture are discussed. 1. Introduction. The ring MU∗(MU) of cooperations in complex cobordism theory has a wellknown interpretation in terms of onedimensional commutative formal group laws. The category C of plocal comodules over MU∗(MU) has a filtration C = C0 ⊃ C1 ⊃···⊃Cn ⊃···