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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
The sigma orientation is an H∞ map
 American Journal of Mathematics
"... Abstract. In [AHS01] the authors constructed a natural map, called the sigma orientation, from the Thom spectrum MU〈6 〉 to any elliptic spectrum in the sense of [Hop95]. MU〈6 〉 is an H ∞ ring spectrum, and in this paper we show that if (E, C, t) is the elliptic spectrum associated to the universal d ..."
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Cited by 15 (2 self)
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Abstract. In [AHS01] the authors constructed a natural map, called the sigma orientation, from the Thom spectrum MU〈6 〉 to any elliptic spectrum in the sense of [Hop95]. MU〈6 〉 is an H ∞ ring spectrum, and in this paper we show that if (E, C, t) is the elliptic spectrum associated to the universal deformation of a supersingular elliptic curve over a perfect field of characteristic p> 0, then the sigma orientation is a map of H ∞ ring spectra.
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 5 (0 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.
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.
A SheafTheoretic View Of Loop Spaces
 Theory Appl. Categ
, 2001
"... The context of enriched sheaf theory introduced in the author's thesis provides a convenient viewpoint for models of the stable homotopy category as well as categories of finite loop spaces. Also, the languages of algebraic geometry and algebraic topology have been interacting quite heavily in re ..."
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Cited by 1 (0 self)
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The context of enriched sheaf theory introduced in the author's thesis provides a convenient viewpoint for models of the stable homotopy category as well as categories of finite loop spaces. Also, the languages of algebraic geometry and algebraic topology have been interacting quite heavily in recent years, primarily due to the work of Voevodsky and that of Hopkins. Thus, the language of Grothendieck topologies is becoming a necessary tool for the algebraic topologist. The current article is intended to give a somewhat relaxed introduction to this language of sheaves in a topological context, using familiar examples such as nfold loop spaces and pointed Gspaces. This language also includes the diagram categories of spectra from [19] as well as spectra in the sense of [17], which will be discussed in some detail. 1.
THE σORIENTATION IS AN H∞ MAP
, 2002
"... Abstract. In [AHS01] the authors introduced the notion of an elliptic spectrum, and constructed a natural map from the Thom spectrum MU〈6 〉 to any elliptic spectrum. MU〈6 〉 is an H ∞ ring spectrum, and in this paper we show that if E is a K(2)local, H ∞ elliptic spectrum, then the σorientation is ..."
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Abstract. In [AHS01] the authors introduced the notion of an elliptic spectrum, and constructed a natural map from the Thom spectrum MU〈6 〉 to any elliptic spectrum. MU〈6 〉 is an H ∞ ring spectrum, and in this paper we show that if E is a K(2)local, H ∞ elliptic spectrum, then the σorientation is a map of H∞ ring spectra.
ATG Topological Hochschild cohomology and
, 2004
"... 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 ..."
Abstract
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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 and Greenlees and of Schwede and 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 EilenbergMac Lane 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.