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26
Algebraic Ktheory of topological Ktheory
"... Let ℓp be the pcomplete connective Adams summand of topological Ktheory, with coefficient ring (ℓp) ∗ = Zp[v1], and let V (1) be the Smith–Toda complex, with BP∗(V (1)) = BP∗/(p, v1). For p ≥ 5 we explicitly compute the V (1)homotopy of the algebraic Ktheory spectrum of ℓp, denoted V (1)∗K(ℓp ..."
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Cited by 21 (10 self)
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Let ℓp be the pcomplete connective Adams summand of topological Ktheory, with coefficient ring (ℓp) ∗ = Zp[v1], and let V (1) be the Smith–Toda complex, with BP∗(V (1)) = BP∗/(p, v1). For p ≥ 5 we explicitly compute the V (1)homotopy of the algebraic Ktheory spectrum of ℓp, denoted V (1)∗K(ℓp). In particular we find that it is a free finitely generated module over the polynomial algebra P (v2), except for a sporadic class in degree 2p − 3. Thus also in this case algebraic Ktheory increases chromatic complexity by one. The proof uses the cyclotomic trace map from algebraic Ktheory to topological cyclic homology, and the calculation is
Topological automorphic forms
 Memoirs of the American Mathematical Society
"... ix 0.1. Background and motivation ix 0.2. Subject matter of this book xvii ..."
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Cited by 18 (7 self)
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ix 0.1. Background and motivation ix 0.2. Subject matter of this book xvii
Moduli Spaces of Commutative Ring Spectra
, 2003
"... Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as ..."
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Cited by 17 (0 self)
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Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as a moduli problem, and give a way  suggested by work of Dwyer, Kan, and Stover  of dissecting the resulting moduli space as a tower with layers governed by appropriate AndreQuillen cohomology groups. A special case is A = E#E itself. The final section applies this to discuss the LubinTate or Morava spectra En .
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
(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 12 (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.
Homotopy fixed points for L K(n)(En∧ X) using the continuous action
 J. Pure Appl. Algebra
"... Abstract. Let G be a closed subgroup of Gn, the extended Morava stabilizer group. Let En be the LubinTate spectrum, X an arbitrary spectrum with trivial Gaction, and let ˆ L = L K(n). We prove that ˆ L(En ∧ X) is a continuous Gspectrum with homotopy fixed point spectrum ( ˆ L(En ∧ X)) hG, define ..."
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Cited by 12 (4 self)
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Abstract. Let G be a closed subgroup of Gn, the extended Morava stabilizer group. Let En be the LubinTate spectrum, X an arbitrary spectrum with trivial Gaction, and let ˆ L = L K(n). We prove that ˆ L(En ∧ X) is a continuous Gspectrum with homotopy fixed point spectrum ( ˆ L(En ∧ X)) hG, defined with respect to the continuous action. Also, we construct a descent spectral sequence whose abutment is π∗( ( ˆ L(En∧X)) hG). We show that the homotopy fixed points of ˆ L(En ∧ X) come from the K(n)localization of the homotopy fixed points of the spectrum (Fn ∧ X). 1.
Moduli problems for structured ring spectra
 DANIEL DUGGER AND BROOKE
, 2005
"... In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theore ..."
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Cited by 10 (0 self)
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In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theorem on the LubinTate or Morava spectra En; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (En)∗X ∼ = (En)∗En as commutative (En) ∗ algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E ∗ and a commutative ring A in E∗E comodules and we wish to discuss the homotopy type of the space T M(A) of all E∞ring spectra so that E∗X ∼ = A. We do not, a priori, assume that T M(A) is nonempty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.
The homotopy fixed point spectra of profinite Galois extensions
 Trans. Amer. Math. Soc
"... Abstract. Let E be a klocal profinite GGalois extension of an E∞ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete Gspectrum. Also, we prove that if E is a profaithful klocal profinite extension which satisfies certain extra conditions, then the for ..."
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Cited by 10 (8 self)
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Abstract. Let E be a klocal profinite GGalois extension of an E∞ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete Gspectrum. Also, we prove that if E is a profaithful klocal profinite extension which satisfies certain extra conditions, then the forward direction of Rognes’s Galois correspondence extends to the profinite setting. We show that the function spectrum FA((E hH)k, (E hK)k) is equivalent to the localized homotopy fixed point spectrum ((E[[G/H]]) hK)k where H and K are closed subgroups of G. Applications to Morava Etheory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action in terms of the derived functor of fixed points.
Γcohomology of rings of numerical polynomials and E∞ structures on
"... Abstract. We investigate Γcohomology of some commutative cooperation algebras E∗E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Γcohomology vanishes above degree 1. As these cohomology groups are the obstruction group ..."
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Cited by 7 (6 self)
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Abstract. We investigate Γcohomology of some commutative cooperation algebras E∗E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Γcohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E ∞ structures. As a consequence we obtain an E∞ structure for the connective Adams summand. For the JohnsonWilson spectrum E(n) with n � 1 we establish the existence of a unique E ∞ structure for its Inadic completion.
Elliptic cohomology
 In preparation
"... This paper is an expository account of the relationship between elliptic cohomology and the emerging subject of derived algebraic geometry. We begin in §1 with an overview of the classical theory of elliptic cohomology. In §2 we review the theory of E∞ring spectra and introduce the language of deri ..."
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This paper is an expository account of the relationship between elliptic cohomology and the emerging subject of derived algebraic geometry. We begin in §1 with an overview of the classical theory of elliptic cohomology. In §2 we review the theory of E∞ring spectra and introduce the language of derived algebraic geometry. We apply this theory in §3, where we introduce the notion of an oriented group scheme and describe connection between oriented group schemes and equivariant cohomology theories. In §4 we sketch a proof of our main result, which relates the classical theory of elliptic cohomology to the classification of oriented elliptic curves. In §5 we discuss various applications of these ideas, many of which rely upon a special feature of elliptic cohomology which we call 2equivariance. The theory that we are going to describe lies at the intersection of homotopy theory and algebraic geometry. We have tried to make our exposition accessible to those who are not specialists in algebraic topology; however, we do assume the reader is familiar with the language of algebraic geometry, particularly with the theory of elliptic curves. In order to keep our account readable, we will gloss over many details, particularly where the use of higher category theory is required. A more comprehensive account of the material described here, with complete definitions and proofs, will be given in [21]. In carrying out the work described in this paper, I have benefitted from the ideas of many people. I