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40
Elliptic spectra, the Witten genus and the theorem of the cube
 Invent. Math
, 1997
"... 2. More detailed results 7 2.1. The algebraic geometry of even periodic ring spectra 7 ..."
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Cited by 96 (18 self)
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2. More detailed results 7 2.1. The algebraic geometry of even periodic ring spectra 7
Comodules and Landweber exact homology theories
 Adv. Math
"... Abstract. We show that, if E is a commutative MUalgebra spectrum such ..."
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Cited by 22 (1 self)
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Abstract. We show that, if E is a commutative MUalgebra spectrum such
(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.
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 15 (8 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.
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
Hochschild cohomology and moduli spaces of strongly homotopy associative algebras
 Homology Homotopy Appl
"... Abstract. Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a twocell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hoc ..."
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Cited by 11 (5 self)
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Abstract. Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a twocell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hochschild cohomology rings of resulting A∞algebras have an interpretation as totally ramified extensions of discrete valuation rings. All A∞algebras are supposed to be unital and we give a detailed analysis of unital structures which is of independent interest. Keywords: A∞algebra, derivation, Hochschild cohomology, formal power series. 1.
Topological Hochschild cohomology and generalized Morita equivalence, Algebraic & Geometric Topology 4
, 2004
"... Abstract. We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case ..."
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Cited by 9 (1 self)
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Abstract. We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case when M is not necessarily a progenerator. Our approach is complementary to recent work of Dwyer & Greenlees and of Schwede & Shipley. A central notion of noncommutative ring theory related to Morita equivalence is that of central separable or Azumaya algebras. For such an Azumaya algebra A, its Hochschild cohomology HH ∗ (A,A) is concentrated in degree 0 and is equal to the center of A. We introduce a notion of topological Azumaya algebra and show that in the case when the ground Salgebra R is an EilenbergMacLane spectrum of a commutative ring this notion specializes to classical Azumaya algebras. A canonical example of a topological Azumaya Ralgebra is the endomorphism Ralgebra FR(M, M) of a finite cell Rmodule. We show that the spectrum of mod 2 topological Ktheory KU/2 is a nontrivial topological Azumaya algebra over the 2adic completion of the Ktheory spectrum ̂ KU2. This leads to the determination of THH(KU/2, KU/2), the topological Hochschild cohomology of KU/2. As far as we know this is the first calculation of THH(A, A) for a noncommutative Salgebra A.