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10
Galois extensions of structured ring spectra
, 2005
"... We introduce the notion of a Galois extension of commutative Salgebras (E ∞ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg–MacLane spectra of commutative rings, real and complex topological Ktheory, Lubin–Tate ..."
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Cited by 19 (1 self)
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We introduce the notion of a Galois extension of commutative Salgebras (E ∞ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg–MacLane spectra of commutative rings, real and complex topological Ktheory, Lubin–Tate spectra and cochain Salgebras. We establish the main theorem of Galois theory in this generality. Its proof involves the notions of separable (and étale) extensions of commutative Salgebras, and the Goerss–Hopkins–Miller theory for E ∞ mapping spaces. We show that the global sphere spectrum S is separably closed (using Minkowski’s discriminant theorem), and we estimate the separable closure of its localization with respect to each of the Morava Ktheories. We also define Hopf–Galois extensions of commutative Salgebras, and study the complex cobordism spectrum MU as a common integral model for all of the local Lubin–Tate Galois extensions.
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.
Localization of AndréQuillenGoodwillie towers, and the periodic homology of infinite loopspaces
, 2003
"... Let K(n) be the n th Morava K–theory at a prime p, and let T(n) be the telescope of a vn–self map of a finite complex of type n. In this paper we study the K(n)∗–homology of Ω ∞ X, the 0 th space of a spectrum X, and many related matters. We give a sampling of our results. Let PX be the free commut ..."
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Cited by 8 (4 self)
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Let K(n) be the n th Morava K–theory at a prime p, and let T(n) be the telescope of a vn–self map of a finite complex of type n. In this paper we study the K(n)∗–homology of Ω ∞ X, the 0 th space of a spectrum X, and many related matters. We give a sampling of our results. Let PX be the free commutative S–algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map sn(X) : LT(n)P(X) → LT(n)Σ ∞ (Ω ∞ X)+ of commutative algebras over the localized sphere spectrum LT(n)S. The induced map of commutative, cocommutative K(n)∗–Hopf algebras sn(X) ∗ : K(n)∗(PX) → K(n)∗(Ω ∞ X), satistfies the following properties. It is always monic. It is an isomorphism if X is n–connected, πn+1(X) is torsion, and T(i)∗(X) = 0 for 1 ≤ i ≤ n−1. It is an isomorphism only if K(i)∗(X) = 0 for 1 ≤ i ≤ n − 1. It is universal: the domain of sn(X) ∗ preserves K(n)∗–isomorphisms, and if F is any functor preserving K(n)∗–isomorphisms, then any natural transformation F(X) → K(n)∗(Ω ∞ X) factors uniquely through sn(X)∗. The construction of our natural transformation uses the telescopic functors constructed and studied previously by Bousfield and the author, and thus depends heavily on the Nilpotence Theorem of Devanitz, Hopkins, and Smith. Our proof that sn(X) ∗ is always monic uses Topological André–Quillen Homology and Goodwillie Calculus in nonconnective settings.
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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Cited by 4 (2 self)
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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
Cohomology theories for highly structured ring spectra. arXiv: math.AT/0211275
"... Abstract. This is a survey paper on cohomology theories for A ∞ and E ∞ ring spectra. Different constructions and main properties of topological AndréQuillen cohomology and of topological derivations are described. We give sample calculations of these cohomology theories and outline applications to ..."
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Cited by 3 (1 self)
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Abstract. This is a survey paper on cohomology theories for A ∞ and E ∞ ring spectra. Different constructions and main properties of topological AndréQuillen cohomology and of topological derivations are described. We give sample calculations of these cohomology theories and outline applications to the existence of A ∞ and E ∞ structures on various spectra. We also explain the relationship between topological derivations, spaces of multiplicative maps and moduli spaces of multiplicative structures. 1.
Topological AndréQuillen homology for cellular commutative
 Salgebras, Abhand. Math. Sem. Univ. Hamburg
"... Abstract. Topological AndréQuillen homology for commutative Salgebras was introduced by Basterra following work of Kriz, and has been intensively studied by several authors. In this paper we discuss it as a homology theory on CW commutative Salgebras and apply it to obtain results on minimal atom ..."
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Cited by 3 (2 self)
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Abstract. Topological AndréQuillen homology for commutative Salgebras was introduced by Basterra following work of Kriz, and has been intensively studied by several authors. In this paper we discuss it as a homology theory on CW commutative Salgebras and apply it to obtain results on minimal atomic plocal Salgebras which generalise those of Baker and May for plocal spectra and simply connected spaces. We exhibit some new examples of minimal atomic Salgebras.
THE BOUSFIELDKUHN FUNCTOR AND TOPOLOGICAL ANDRÉQUILLEN COHOMOLOGY
"... 2. Recollections on the DyerLashof algebra for Morava Etheory 3 3. BarrBeck homology 8 4. Topological AndréQuillen homology 11 ..."
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Cited by 2 (0 self)
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2. Recollections on the DyerLashof algebra for Morava Etheory 3 3. BarrBeck homology 8 4. Topological AndréQuillen homology 11
MODEL CATEGORY EXTENSIONS OF THE PIRASHVILISLOMIŃSKA THEOREMS
, 806
"... Abstract. We describe the class of semistable model categories, which generalize the equivalence of finite products and coproducts in abelian and stable model categories, and use this to establish Morita equivalences among categories of functors. We provide a construction of pairs of small categori ..."
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Abstract. We describe the class of semistable model categories, which generalize the equivalence of finite products and coproducts in abelian and stable model categories, and use this to establish Morita equivalences among categories of functors. We provide a construction of pairs of small categories—known as conjugate pairs—whose associated categories of diagrams are Quillen equivalent in the semistable setting. We frame our development in the context of Morita theory, following Slomińska’s work on similar questions for categories of functors enriched over and taking values in Rmodules.