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25
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 29 (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.
Power operations in elliptic cohomology and representations of loop groups
 Trans. Amer. Math. Soc
, 2000
"... Abstract. The first part describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve. The second part discusses a relationship between equivariant elliptic cohomology and representations of loop groups. The third part investigates the representation theor ..."
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Cited by 23 (5 self)
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Abstract. The first part describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve. The second part discusses a relationship between equivariant elliptic cohomology and representations of loop groups. The third part investigates the representation theoretic considerations which give rise to the power operations discussed in the first part. Contents
Varieties and local cohomology for chromatic group cohomology rings
 Topology
, 1999
"... where E is a suitable complete periodic complex oriented theory and G is a finite group: we describe its variety in terms of the formal group associated to E, and the category of abelian psubgroups of G. Our results considerably extend those of HopkinsKuhnRavenel [16], and this enables us to obta ..."
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Cited by 19 (11 self)
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where E is a suitable complete periodic complex oriented theory and G is a finite group: we describe its variety in terms of the formal group associated to E, and the category of abelian psubgroups of G. Our results considerably extend those of HopkinsKuhnRavenel [16], and this enables us to obtain information about the associated homology of BG. For example if E is the complete 2periodic version of the JohnsonWilson theory E(n) the irreducible components of the variety of the quotient E (BG)=I k by the invariant prime ideal I k = (p; v 1; : : : ; v k\Gamma1) correspond to conjugacy classes of abelian psubgroups of rank n \Gamma k. Furthermore, if we invert v k the decomposition of the variety into irreducible pieces corresponding to minimal primes becomes a decomposition
(Pre)sheaves of Ring Spectra over the Moduli Stack of Formal Group Laws
, 2004
"... In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem. ..."
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Cited by 17 (1 self)
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In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem.
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
GrossHopkins duality
 Topology
"... In [8] Hopkins and Gross state a theorem revealing a profound relationship between two different kinds of duality in stable homotopy theory. A proof of a related but weaker result is given in [3], and we understand that Sadofsky is preparing a proof that works in general. Here we present a proof tha ..."
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Cited by 9 (0 self)
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In [8] Hopkins and Gross state a theorem revealing a profound relationship between two different kinds of duality in stable homotopy theory. A proof of a related but weaker result is given in [3], and we understand that Sadofsky is preparing a proof that works in general. Here we present a proof that seems rather different and complementary to Sadofsky’s. We thank IChiau Huang for help with Proposition 18, and John Greenlees for helpful discussions. We first indicate the context of the HopkinsGross theorem. Cohomological duality theorems have been studied in a number of contexts; they typically say that H k (X ∗ ) = H d−k (X) ∨ for some class of objects X with some notion of duality X ↔ X ∗ and some type of cohomology groups H k (X) with some notion of duality A ↔ A ∨ and some integer d. For example, if M is a compact smooth oriented manifold of dimension d we have a Poincaré duality isomorphism H k (M; Q) = Hom(H d−k (M; Q), Q) (so here we just have M ∗ = M). For another example, let S be a smooth complex projective variety of dimension d, and let Ω d be the sheaf of topdimensional differential forms. Then for any coherent sheaf F on S we have a Serre duality isomorphism H k (S; Hom(F, Ω d)) = Hom(H d−k (S; F), C). This can be seen as a special case of the Grothendieck duality theorem for a proper morphism [7], which is formulated in terms of functors between derived categories. There is a wellknown analogy
On degeneration of onedimensional formal group laws and stable homotopy theory
, 2003
"... 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 gen ..."
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Cited by 7 (1 self)
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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.
Discrete torsion for the supersingular orbifold sigma genus
, 2003
"... The first purpose of this paper is to examine the relationship between equivariant elliptic genera and orbifold elliptic genera. We apply the character theory of [HKR00] to the Borelequivariant genus associated to the sigma orientation of [AHS01] to define an orbifold genus for certain total quoti ..."
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Cited by 5 (0 self)
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The first purpose of this paper is to examine the relationship between equivariant elliptic genera and orbifold elliptic genera. We apply the character theory of [HKR00] to the Borelequivariant genus associated to the sigma orientation of [AHS01] to define an orbifold genus for certain total quotient orbifolds and supersingular elliptic curves. We show that our orbifold genus is given by the same sort of formula as the orbifold “twovariable ” genus of [DMVV97] and [BL02]. In the case of a finite cyclic orbifold group, we use the characteristic series for the twovariable genus in the formulae of [And03] to define an analytic equivariant genus in Grojnowski’s equivariant elliptic cohomology, and we show that this gives precisely the orbifold twovariable genus. The second purpose of this paper is to study the effect of varying the BU〈6〉structure in the Borelequivariant sigma orientation. We show that varying the BU〈6 〉 structure by a class in H 3 (BG; Z), where G is the orbifold group, produces discrete torsion in the sense of [Vaf85]. This result was first obtained by Sharpe [Sha], for a different orbifold genus and using different methods.
THE CONGRUENCE CRITERION FOR POWER OPERATIONS IN MORAVA ETHEORY
, 2009
"... We prove a congruence criterion for the algebraic theory of power operations in Morava Etheory, analogous to Wilkerson’s congruence criterion for torsion free λrings. In addition, we provide a geometric description of this congruence criterion, in terms of sheaves on the moduli problem of deform ..."
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Cited by 4 (1 self)
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We prove a congruence criterion for the algebraic theory of power operations in Morava Etheory, analogous to Wilkerson’s congruence criterion for torsion free λrings. In addition, we provide a geometric description of this congruence criterion, in terms of sheaves on the moduli problem of deformations of formal groups and Frobenius isogenies.