A filtration in algebraic K-theory

Duflot, Jeanne; Marak, C.Tyrel

doi:10.1016/s0022-4049(99)00050-x

Cited by 3 publications

(15 citation statements)

References 6 publications

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“…where J, B, C are homomorphisms described in [7], and δ 2 , δ 1 are connecting homomorphisms arising in exact sequences associated to certain nonabelian cohomology groups. The pertinent definitions are reviewed below.…”

Section: Connections With Classical K-theorymentioning

confidence: 99%

“…This paper is the initial part of a study of the Bousfield-Kan spectral sequence associated to certain types of cosimplicial simplicial groups and its relationship to the work presented in [6,7]. These relationships are primarily developed in Sections 5 and 6.…”

Section: Introductionmentioning

confidence: 99%

“…• Sections 4 through 5.2 provide a review of notation, definitions and results from [6,7]. In [7], if R is the inverse limit of a diagram of rings R over a small category I, then a filtration…”

Section: Introductionmentioning

confidence: 99%

“…In [7], if R is the inverse limit of a diagram of rings R over a small category I, then a filtration…”

Section: Introductionmentioning

confidence: 99%

“…Classically, this was initiated by Milnor [13]. The main result of this paper is the direct comparison of the filtration in classical algebraic K-theory discussed in [6,7] to a corresponding filtration in the BousfieldKan spectral sequence associated to a Tot-tower of simplicial groups attached to the diagram of rings. …”

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confidence: 98%

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The algebraic $K$-theory of a diagram of rings

Duflot

2008

Homology, Homotopy and Applications

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In this paper, we consider "diagrams of rings", or functors from a small category to the category of rings, and the corresponding diagrams of groups K i . Classically, this was initiated by Milnor [13]. The main result of this paper is the direct comparison of the filtration in classical algebraic K-theory discussed in [6,7] to a corresponding filtration in the BousfieldKan spectral sequence associated to a Tot-tower of simplicial groups attached to the diagram of rings.

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Section: Connections With Classical K-theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In [7], if R is the inverse limit of a diagram of rings R over a small category I, then a filtration…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 98%

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The algebraic $K$-theory of a diagram of rings

Duflot

2008

Homology, Homotopy and Applications

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Second cohomotopy and nonabelian cohomology

Pirashvili¹

2014

J. K-Theory

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The main difficulty in the theory of non-abelian cohomology is that for cosimplicial groups only zero-th and first dimensional cohomotopy are known. In this article we introduce a new class of cosimplicial groups, called centralised cosimplicial groups, for which we are able to define a second cohomotopy, with all expected properties. The main examples of such cosimplicial groups come from 2-categories.

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The motivic anabelian geometry of local heights on abelian varieties

Betts¹

2017

Preprint

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We study the problem of describing local components of height functions on abelian varieties over characteristic 0 local fields as functions on spaces of torsors under various realisations of a 2-step unipotent motivic fundamental group naturally associated to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the Q ℓ -and Qp-pro-unipotent étale realisations when the base field is p-adic, and in terms of the R-pro-unipotent Betti-de Rham realisation when the base field is archimedean.In the course of proving the p-adic instance of these theorems, we develop a new technique for studying local non-abelian Bloch-Kato Selmer sets, working with certain explicit cosimplicial group models for these sets and using methods from homotopical algebra. Among other uses, these models enable us to construct a non-abelian generalisation of the Bloch-Kato exponential sequence under minimal conditions.On the geometric side, we also prove a number of foundational results on local constancy or analyticity of various non-abelian Kummer maps for prounipotent étale or Betti-de Rham fundamental groups of arbitrary smooth varieties.

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A filtration in algebraic K-theory

Cited by 3 publications

References 6 publications

The algebraic $K$-theory of a diagram of rings

The algebraic $K$-theory of a diagram of rings

Second cohomotopy and nonabelian cohomology

The motivic anabelian geometry of local heights on abelian varieties

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