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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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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 .
(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.
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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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.
Realizability of algebraic Galois extensions by strictly commutative ring spectra
"... Abstract. We discuss some of the basic ideas of Galois theory for commutative Salgebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central rôles are ..."
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Cited by 9 (6 self)
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Abstract. We discuss some of the basic ideas of Galois theory for commutative Salgebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones applying obstruction theories of Robinson and GoerssHopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative Salgebras. Examples such as the complex Ktheory spectrum as a KOalgebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones and this leads to computable Harrison groups for such spectra. We end by proving an analogue of Hilbert’s theorem 90 for the units associated with a Galois extension.
Iterated bar complexes of Einfinity algebras and homology theories
, 2008
"... We proved in a previous article that the bar complex of an E ∞algebra inherits a natural E ∞algebra structure. As a consequence, a welldefined iterated bar construction B n (A) can be associated to any algebra over an E ∞operad. In the case of a commutative algebra A, our iterated bar constructi ..."
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Cited by 5 (2 self)
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We proved in a previous article that the bar complex of an E ∞algebra inherits a natural E ∞algebra structure. As a consequence, a welldefined iterated bar construction B n (A) can be associated to any algebra over an E ∞operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E ∞algebras. We use this effective definition to prove that the nfold bar construction admits an extension to categories of algebras over Enoperads. Then we prove that the nfold bar complex determines the homology theory associated to the category of algebras over an Enoperad. In the case n = ∞, we obtain an isomorphism between the homology of an infinite bar construction and the usual Γhomology with trivial coefficients.
Co)homology theories for commutative Salgebras
"... The aim of this paper is to give an overview of some of the existing homology theories for commutative (S)algebras. We do not claim any originality; nor do we pretend to give a complete account. But the results in that field are widely spread in the literature, so for someone who does not actually ..."
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The aim of this paper is to give an overview of some of the existing homology theories for commutative (S)algebras. We do not claim any originality; nor do we pretend to give a complete account. But the results in that field are widely spread in the literature, so for someone who does not actually work in that subject, it can be difficult to trace all the relationships between the different homology theories. The theories we aim to compare are • topological AndréQuillen homology • Gamma homology • stable homotopy of Γmodules • stable homotopy of algebraic theories • the AndréQuillen cohomology groups which arise as obstruction groups in the GoerssHopkins approach As a comparison between stable homotopy of Γmodules and stable homotopy of algebraic theories is not explicitly given in the literature, we will give a proof of Theorem 2.1 which says that both homotopy theories are isomorphic
Iterated bar complexes of E∞ algebras and homology theories
, 2010
"... We proved in a previous article that the bar complex of an E∞algebra inherits a natural E∞algebra structure. As a consequence, a welldefined iterated bar construction Bn (A) can be associated to any algebra over an E∞operad. In the case of a commutative algebra A, our iterated bar construction re ..."
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We proved in a previous article that the bar complex of an E∞algebra inherits a natural E∞algebra structure. As a consequence, a welldefined iterated bar construction Bn (A) can be associated to any algebra over an E∞operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E∞algebras. We use this effective definition to prove that the nfold bar construction admits an extension to categories of algebras over Enoperads. Then we prove that the nfold bar complex determines the homology theory associated to the category of algebras over an Enoperad. In the case n = ∞, we obtain an isomorphism between the homology of an infinite bar construction and the usual Γhomology with trivial coefficients.
THE COLLAPSE OF THE PERIODICITY SEQUENCE IN THE STABLE RANGE
, 2006
"... Abstract. The stabilization of Hochschild homology of commutative algebras is Gamma homology. We describe a cyclic variant of Gamma homology and prove that the associated analogue of Connes ’ periodicity sequence becomes almost trivial, because the cyclic version coincides with the ordinary version ..."
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Abstract. The stabilization of Hochschild homology of commutative algebras is Gamma homology. We describe a cyclic variant of Gamma homology and prove that the associated analogue of Connes ’ periodicity sequence becomes almost trivial, because the cyclic version coincides with the ordinary version from homological degree two on. We offer an alternative explanation for this by proving that the Boperator followed by the stabilization map is trivial from degree one on. 1.
UNIQUENESS OF E ∞ STRUCTURES FOR CONNECTIVE COVERS
, 2006
"... Abstract. We refine our earlier work on the existence and uniqueness of E ∞ structures on Ktheoretic spectra to show that at each prime p, the connective Adams summand ℓ has a unique structure as a commutative Salgebra. For the pcompletion ℓp we show that the McClureStaffeldt model for ℓp is equ ..."
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Abstract. We refine our earlier work on the existence and uniqueness of E ∞ structures on Ktheoretic spectra to show that at each prime p, the connective Adams summand ℓ has a unique structure as a commutative Salgebra. For the pcompletion ℓp we show that the McClureStaffeldt model for ℓp is equivalent as an E ∞ ring spectrum to the connective cover of the periodic Adams summand Lp. We establish a Bousfield equivalence between the connective cover of the LubinTate spectrum En and BP〈n〉.
unknown title
"... arXiv version: fonts, pagination and layout may vary from AGT published version A lower bound for coherences on the Brown–Peterson spectrum ..."
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arXiv version: fonts, pagination and layout may vary from AGT published version A lower bound for coherences on the Brown–Peterson spectrum