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Modern foundations of stable homotopy theory. Handbook of Algebraic Topology, edited by
, 1995
"... 2. Smash products and twisted halfsmash products 11 ..."
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Cited by 22 (7 self)
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2. Smash products and twisted halfsmash products 11
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 12 (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.
Hopf algebra structure on topological Hochschild homology
, 2005
"... The topological Hochschild homology THH(R) of a commutative Salgebra (E ∞ ring spectrum) R naturally has the structure of a Hopf algebra over R, in the homotopy category. We show that under a flatness assumption this makes the Bökstedt spectral sequence converging to the mod p homology of THH(R) in ..."
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Cited by 6 (3 self)
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The topological Hochschild homology THH(R) of a commutative Salgebra (E ∞ ring spectrum) R naturally has the structure of a Hopf algebra over R, in the homotopy category. We show that under a flatness assumption this makes the Bökstedt spectral sequence converging to the mod p homology of THH(R) into a Hopf algebra spectral sequence. We then apply this additional structure to study some interesting examples, including the commutative Salgebras ku, ko, tmf, ju and j, and to calculate the homotopy groups of THH(ku) and THH(ko) after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic Ktheory of Salgebras, using topological cyclic homology.
The units of a ring spectrum and a logarithmic cohomology operation
 J. Amer. Math. Soc
"... Recall that if R is a commutative ring, then the set R × ⊂ R of invertible elements of R is naturally an abelian group under multiplication. This construction is a functor from commutative rings to abelian groups. In general, there is no obvious relation between the additive group of a ring R and t ..."
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Cited by 6 (0 self)
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Recall that if R is a commutative ring, then the set R × ⊂ R of invertible elements of R is naturally an abelian group under multiplication. This construction is a functor from commutative rings to abelian groups. In general, there is no obvious relation between the additive group of a ring R and the multiplicative group of
Orbifold genera, product formulas and power operations
 Adv. Math
, 2006
"... Moore, Verlinde and Verlinde, expressing the orbifold elliptic genus of the symmetric powers of an almost complex manifold M in terms of the elliptic genus of M itself. We show that from the point of view of elliptic cohomology an analogous ptypical statement follows as an easy corollary from the f ..."
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Cited by 5 (3 self)
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Moore, Verlinde and Verlinde, expressing the orbifold elliptic genus of the symmetric powers of an almost complex manifold M in terms of the elliptic genus of M itself. We show that from the point of view of elliptic cohomology an analogous ptypical statement follows as an easy corollary from the fact that the map of spectra corresponding to the genus preserves power operations. We define higher chromatic versions of the notion of orbifold genus, involving htuples rather than pairs of commuting elements. Using homotopy theoretic methods we are able to prove an integrality result and show that our definition is independent of the representation of the orbifold. Our setup is so simple, that it allows us to prove DMVVtype product formulas for these higher chromatic orbifold genera in the same way that the product formula for the topological Todd genus is proved. More precisely, we show that any genus induced by an H∞map into one of the MoravaLubinTate cohomology theories Eh has such a product formula and that the formula depends only on h and not on the genus. Since the complex H∞genera into Eh have been classified in [And95], a large family of genera to which our results apply is completely understood. Loosely speaking, our result says that some Borcherds
Tate cohomology and periodic localization of polynomial functors
 Invent. Math
"... Abstract. In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a vn self map of a finite S–module of type n. The Per ..."
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Cited by 4 (2 self)
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Abstract. In this paper, we show that Goodwillie calculus, as applied to functors from stable homotopy to itself, interacts in striking ways with chromatic aspects of the stable category. Localized at a fixed prime p, let T(n) be the telescope of a vn self map of a finite S–module of type n. The Periodicity Theorem of Hopkins and Smith implies that the Bousfield localization functor associated to T(n) ∗ is independent of choices. Goodwillie’s general theory says that to any homotopy functor F from S–modules to S–modules, there is an associated tower under F, {PdF}, such that F → PdF is the universal arrow to a d–excisive functor. Our first main theorem says that PdF → Pd−1F always admits a homotopy section after localization with respect to T(n) ∗ (and so also after localization with respect to Morava K–theory K(n)∗). Thus, after periodic localization, polynomial functors split as the product of their homogeneous factors. This theorem follows from our second main theorem which is equivalent to the following: for any finite group G, the Tate spectrum tG(T(n)) is weakly contractible. This strengthens and extends previous theorems of Greenlees–Sadofsky, Hovey–Sadofsky, and Mahowald–Shick. The Periodicity Theorem is used in an essential way in our proof. The connection between the two theorems is via a reformulation of a result of McCarthy on dual calculus.
CORES OF SPACES, SPECTRA, AND E ∞ RING SPECTRA
"... Abstract. In a paper that has attracted little notice, Priddy showed that the BrownPeterson spectrum at a prime p can be constructed from the plocal sphere spectrum S by successively killing its odd dimensional homotopy groups. This seems to be an isolated curiosity, but it is not. For any space o ..."
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Cited by 3 (0 self)
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Abstract. In a paper that has attracted little notice, Priddy showed that the BrownPeterson spectrum at a prime p can be constructed from the plocal sphere spectrum S by successively killing its odd dimensional homotopy groups. This seems to be an isolated curiosity, but it is not. For any space or spectrum Y that is plocal and (n0 − 1)connected and has πn0 (Y) cyclic, there is a plocal, (n0 − 1)connected “nuclear ” CW complex or CW spectrum X and a map f: X → Y that induces an isomorphism on πn0 and a monomorphism on all homotopy groups. Nuclear complexes are atomic: a selfmap that induces an isomorphism on πn0 must be an equivalence. The construction of X from Y is neither functorial nor even unique up to equivalence, but it is there. Applied to the localization of MU at p, the construction yields BP. In 1999, the third author gave an April Fool’s talk on how to prove that BP is an E ∞ ring spectrum or, in modern language, a commutative Salgebra. As explained in [20], he gave a quite different April Fool’s talk on the same subject two years earlier. His new idea was to exploit the remarkable paper of Stewart Priddy [23], in which Priddy constructed BP by killing the odd degree homotopy groups of the sphere spectrum. The hope was that by mimicking Priddy’s construction in the category of commutative Salgebras, one might arrive at a construction of BP as a commutative Salgebra. As the first two authors discovered, that argument fails. However, the ideas are still interesting. As we shall explain, Priddy’s construction of BP is not an accidental fluke but rather a special case of a very general construction. The elementary space and spectrum level construction is given in Section 1. The more sophisticated E ∞ ring spectrum analogue and its specialization to MU are discussed in Section 2. It is a pleasure to thank Nick Kuhn and Fred Cohen for very illuminating emails. In particular, Example 1.10 is due to Cohen. 1. Cores of spaces and spectra
Differentials in the homological homotopy fixed point spectral sequence
, 2005
"... Abstract We analyze in homological terms the homotopy fixed point spectrum of a Tequivariant commutative Salgebra R. There is a homological homotopy fixed point spectral sequence with E2 s,t = H−s gp (T; Ht(R; Fp)), converging conditionally to the continuous homology H c s+t (RhT; Fp) of the homot ..."
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Cited by 2 (0 self)
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Abstract We analyze in homological terms the homotopy fixed point spectrum of a Tequivariant commutative Salgebra R. There is a homological homotopy fixed point spectral sequence with E2 s,t = H−s gp (T; Ht(R; Fp)), converging conditionally to the continuous homology H c s+t (RhT; Fp) of the homotopy fixed point spectrum. We show that there are Dyer–Lashof operations β ǫ Q i acting on this algebra spectral sequence, and that its differentials are completely determined by those originating on the vertical axis. More surprisingly, we show that for each class x in the E 2rterm of the spectral sequence there are 2r other classes in the E 2rterm (obtained mostly by Dyer–Lashof operations on x) that are infinite cycles, i.e., survive to the E ∞term. We apply this to completely determine the differentials in the homological homotopy fixed point spectral sequences for the topological Hochschild homology spectra R = THH(B) of many Salgebras, including B = MU, BP, ku, ko and tmf. Similar results apply for all finite subgroups C ⊂ T, and for the Tate and homotopy orbit spectral sequences. This work is part of a homological approach to calculating topological cyclic homology and algebraic Ktheory of commutative Salgebras.