Regulators and cycle maps in higher-dimensional differential algebraic K-theory

Bunke, Ulrich; Tamme, Georg

doi:10.1016/j.aim.2015.08.004

Cited by 7 publications

(25 citation statements)

References 34 publications

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“…It is our philosophy that algebraic K-theory classes on a smooth variety X are realized in terms of vector bundles parametrized by auxiliary smooth manifolds, and that regulator classes are represented by characteristic forms associated to geometric structures on these bundles, see e.g. [BT15]. In general, the complex E(T ⊗ A log (X)) is too small to contain these characteristic forms.…”

Section: The Decomposition (15) Gives a Decomposition Of Chain Complexesmentioning

confidence: 99%

The Beilinson regulator is a map of ring spectra

Bunke

Nikolaus

Tamme

2018

Advances in Mathematics

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We prove that the Beilinson regulator, which is a map from K-theory to absolute Hodge cohomology of a smooth variety, admits a refinement to a map of E ∞ -ring spectra in the sense of algebraic topology. To this end we exhibit absolute Hodge cohomology as the cohomology of a commutative differential graded algebra over R.The associated spectrum to this CDGA is the target of the refinement of the regulator and the usual K-theory spectrum is the source. To prove this result we compute the space of maps from the motivic K-theory spectrum to the motivic spectrum that represents absolute Hodge cohomology using the motivic Snaith theorem. We identify those maps which admit an E ∞ -refinement and prove a uniqueness result for these refinements. *

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Section: The Decomposition (15) Gives a Decomposition Of Chain Complexesmentioning

confidence: 99%

The Beilinson regulator is a map of ring spectra

Bunke

Nikolaus

Tamme

2018

Advances in Mathematics

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“…It is an isomorphism if X is also proper over C. If one moreover introduces the weight filtrationŴ and replaces A R,log , A log by the subcomplexesŴ 2p A R,log ,Ŵ 2p A log , one obtains the absolute Hodge cohomology H * AH (X, R(p)) introduced by Beilinson [Beȋ86]. This is the cohomology theory used in [BT15]. It follows from Deligne's theory of weights that the natural map H * AH (X, R(p)) → H * BD (X, R(p)) is an isomorphism in degrees * ≤ p, and in degrees * ≤ 2p if X is proper.…”

Section: The Multiplicative Deligne Complexmentioning

confidence: 99%

“…We form the connection ∇ := ∇ I + ∇ II and let ∇ u be its unitarization with respect to h V . In [BT15] we use these connections in order to define a characteristic form in DR(M × X). In the present paper we adjust the notion of a geometry such that we obtain a lift of the characteristic form to IDR(M × X), see Lemma 2.22.…”

Section: Geometries and Characteristic Formsmentioning

confidence: 99%

“…In general, the spectrum K(X) represents a generalized cohomology theory and, for a manifold M , we have π * (K(M × X)) ∼ = K(X) − * (M ) (see [BT15,Section 4.5]).…”

Section: The Multiplicative K-theory Sheaf and The Regulatormentioning

confidence: 99%

“…An important, but extremely difficult problem is to construct K-theory classes and to compute their images under the regulator map. The papers [BG13,BT15] initiated a new approach to this problem. The idea is to represent algebraic K-theory classes of X by bundles on M × X for smooth manifolds M .…”

Section: Introductionmentioning

confidence: 99%

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Multiplicative differential algebraic K-theory and applications

Bunke

Tamme

2016

AKT

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We construct a version of Beilinson's regulator as a map of sheaves of commutative ring spectra and use it to define a multiplicative variant of differential algebraic K-theory. We use this theory to give an interpretation of Bloch's construction of K 3 -classes and the relation with dilogarithms. Furthermore, we provide a relation to Arakelov theory via the arithmetic degree of metrized line bundles, and we give a proof of the formality of the algebraic K-theory of number rings.

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$$G $$-theory of $$\mathbb F_1$$-algebras I: the equivariant Nishida problem

Mahanta

2017

J. Homotopy Relat. Struct.

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Abstract. We develop a version of G-theory for an F 1 -algebra (i.e., the K-theory of pointed G-sets for a pointed monoid G) and establish its first properties. We construct a Cartan assembly map to compare the Chu-Morava K-theory for finite pointed groups with our G-theory. We compute the G-theory groups for finite pointed groups in terms of stable homotopy of some classifying spaces. We introduce certain Loday-Whitehead groups over F 1 that admit functorial maps into classical Whitehead groups under some reasonable hypotheses. We initiate a conjectural formalism using combinatorial Grayson operations to address the Equivariant Nishida Problem -it asks whether S G admits operations that endow ⊕ n π 2n (S G ) with a pre-λ-ring structure, where G is a finite group and S G is the G-fixed point spectrum of the equivariant sphere spectrum.

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Regulators and cycle maps in higher-dimensional differential algebraic K-theory

Cited by 7 publications

References 34 publications

The Beilinson regulator is a map of ring spectra

The Beilinson regulator is a map of ring spectra

Multiplicative differential algebraic K-theory and applications

$$G $$-theory of $$\mathbb F_1$$-algebras I: the equivariant Nishida problem

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