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55
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 24 (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
Localization theorems in topological Hochschild homology and topological cyclic homology
, 2008
"... We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of ..."
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Cited by 22 (3 self)
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We construct localization cofiber sequences for the topological Hochschild homology (THH) and topological cyclic homology (TC) of spectral categories. Using a “global ” construction of the THH and TC of a scheme in terms of the perfect complexes in a spectrally enriched version of the category of unbounded complexes, the sequences specialize to localization cofiber sequences associated to the inclusion of an open subscheme. These are the targets of the cyclotomic trace from the localization sequence of ThomasonTrobaugh in Ktheory. We also deduce versions of Thomason’s blowup formula and the projective bundle formula for THH and TC.
Cyclic polytopes and the Ktheory of truncated polynomial algebras
 Inv. Math
"... This paper calculates the relative algebraic Ktheory K∗(k[x]/(xn), (x)) of a truncated polynomial algebra over a perfect field k of positive characteristic p. Since the ideal generated by x is nilpotent, we can apply McCarthy’s theorem: the relative algebraic Ktheory is isomorphic to the relative ..."
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Cited by 16 (6 self)
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This paper calculates the relative algebraic Ktheory K∗(k[x]/(xn), (x)) of a truncated polynomial algebra over a perfect field k of positive characteristic p. Since the ideal generated by x is nilpotent, we can apply McCarthy’s theorem: the relative algebraic Ktheory is isomorphic to the relative topological cyclic homology, [Mc], and it is the latter groups we actually evaluate.
Topological equivalences for differential graded algebras
 Adv. Math
, 2006
"... Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an EilenbergMac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are ..."
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Cited by 14 (6 self)
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Abstract. We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an EilenbergMac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasiisomorphic dgas are topologically equivalent, but we produce explicit counterexamples of the converse. We also develop an associated notion of topological Morita equivalence using a homotopical version of tilting. Contents
TRACE MAPS FROM THE ALGEBRAIC KTHEORY OF THE INTEGERS (After Marcel Bökstedt)
"... Let p be any prime. We consider Bökstedt’s topological refinement K(Z) → T (Z) = T HH(Z) of the Dennis trace map from algebraic Ktheory of the integers to topological Hochschild homology of the integers. This trace map is shown to induce a surjection on homotopy in degree 2p − 1, onto the first p ..."
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Cited by 10 (9 self)
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Let p be any prime. We consider Bökstedt’s topological refinement K(Z) → T (Z) = T HH(Z) of the Dennis trace map from algebraic Ktheory of the integers to topological Hochschild homology of the integers. This trace map is shown to induce a surjection on homotopy in degree 2p − 1, onto the first ptorsion in the target. Furthermore, Bökstedt’s map factors through the S 1homotopy fixed points T (Z) hS1 of T (Z), and it is shown that the first ptorsion element in degree 2p − 3 of the stable homotopy groups of spheres is detected in the homotopy of T (Z) hS1 Both results are due to Bökstedt, but have remained unpublished.
Topological cyclic homology of schemes
 Preprint 1997 28 THOMAS GEISSER AND MARC LEVINE
, 2001
"... In recent years, the topological cyclic homology functor of [4] has been used to study and to calculate higher algebraic Ktheory. It is known that for finite algebras over the ring of Witt vectors of a perfect field of characteristic p, the padic Ktheory and topological cyclic homology agree in n ..."
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Cited by 10 (0 self)
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In recent years, the topological cyclic homology functor of [4] has been used to study and to calculate higher algebraic Ktheory. It is known that for finite algebras over the ring of Witt vectors of a perfect field of characteristic p, the padic Ktheory and topological cyclic homology agree in nonnegative degrees, [20]. This has been
Topological Hochschild homology of connective complex Ktheory
 Amer. J. Math
"... Abstract. Let ku be the connective complex Ktheory spectrum, completed at an odd prime p. We present a computation of the mod (p, v1) homotopy algebra of the topological Hochschild homology spectrum of ku. 1. Introduction. Since ..."
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Cited by 9 (0 self)
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Abstract. Let ku be the connective complex Ktheory spectrum, completed at an odd prime p. We present a computation of the mod (p, v1) homotopy algebra of the topological Hochschild homology spectrum of ku. 1. Introduction. Since
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 9 (5 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 smooth Whitehead spectrum of a point at odd regular primes
 TOPOL
, 2003
"... Let p be an odd regular prime, and assume that the Lichtenbaum– Quillen conjecture holds for K(Z[1/p]) at p. Then the pprimary homotopy type of the smooth Whitehead spectrum W h(∗) is described. A suspended copy of the cokernelofJ spectrum splits off, and the torsion homotopy of the remainder eq ..."
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Cited by 9 (4 self)
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Let p be an odd regular prime, and assume that the Lichtenbaum– Quillen conjecture holds for K(Z[1/p]) at p. Then the pprimary homotopy type of the smooth Whitehead spectrum W h(∗) is described. A suspended copy of the cokernelofJ spectrum splits off, and the torsion homotopy of the remainder equals the torsion homotopy of the fiber of the restricted S 1transfer map t: ΣCP ∞ → S. The homotopy of W h(∗) is determined in a range of degrees, and the cohomology of W h(∗) is expressed as an Amodule in all degrees, up to an extension. These results have geometric topological interpretations, in terms of spaces of concordances or diffeomorphisms of highly connected, high dimensional compact smooth manifolds.