scholarly journals Rost nilpotence and étale motivic cohomology

2018 ◽  
Vol 330 ◽  
pp. 420-432 ◽  
Author(s):  
Andreas Rosenschon ◽  
Anand Sawant
Keyword(s):  
Motivic Cohomology

Motives and modules over motivic cohomology

2006 ◽  
Vol 342 (10) ◽  
pp. 751-754 ◽  
Author(s):  
Oliver Röndigs ◽  
Paul Arne Østvær
Keyword(s):  
Motivic Cohomology

scholarly journals Néron models and limits of Abel–Jacobi mappings

2010 ◽  
Vol 146 (2) ◽  
pp. 288-366 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Normal Function ◽  
Motivic Cohomology ◽  
Normal Functions ◽  
Geometric Setting ◽  
Neron Models ◽  
A Finite Group ◽  
Intermediate Jacobian ◽  
Néron Models ◽  
Singular Fibre ◽  
Finite Singularity

AbstractWe show that the limit of a one-parameter admissible normal function with no singularities lies in a non-classical sub-object of the limiting intermediate Jacobian. Using this, we construct a Hausdorff slit analytic space, with complex Lie group fibres, which ‘graphs’ such normal functions. For singular normal functions, an extension of the sub-object by a finite group leads to the Néron models. When the normal function comes from geometry, that is, a family of algebraic cycles on a semistably degenerating family of varieties, its limit may be interpreted via the Abel–Jacobi map on motivic cohomology of the singular fibre, hence via regulators onK-groups of its substrata. Two examples are worked out in detail, for families of 1-cycles on CY and abelian 3-folds, where this produces interesting arithmetic constraints on such limits. We also show how to compute the finite ‘singularity group’ in the geometric setting.

Motivic cohomology of quadrics and the coniveau spectral sequence

2010 ◽  
Vol 6 (3) ◽  
pp. 547-589 ◽  
Author(s):  
Nobuaki Yagita
Keyword(s):  
Spectral Sequence ◽  
Spectral Sequences ◽  
Motivic Cohomology

AbstractWe study the coniveau spectral sequence for quadrics defined by Pfister forms. In particular, we explicitly compute the motivic cohomology of anisotropic quadrics over ℝ, by showing that their coniveau spectral sequences collapse from the -term

An Overview of the Proof

2019 ◽  
pp. 3-22
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Motivic Cohomology ◽  
Norm Residue ◽  
Cohomology Operations ◽  
Definition Of ◽  
Special Case

This chapter provides the main steps in the proof of Theorems A and B regarding the norm residue homomorphism. It also proves several equivalent (but more technical) assertions in order to prove the theorems in question. This chapter also supplements its approach by defining the Beilinson–Lichtenbaum condition. It thus begins with the first reductions, the first of which is a special case of the transfer argument. From there, the chapter presents the proof that the norm residue is an isomorphism. The definition of norm varieties and Rost varieties are also given some attention. The chapter also constructs a simplicial scheme and introduces some features of its cohomology. To conclude, the chapter discusses another fundamental tool—motivic cohomology operations—as well as some historical notes.

scholarly journals Algebraic K-Theory and Motivic Cohomology

2009 ◽  
pp. 1731-1774 ◽  
Author(s):  
Thomas Geisser ◽  
Annette Huber-Klawitter ◽  
Uwe Jannsen ◽  
Marc Levine
Keyword(s):  
Motivic Cohomology ◽  
K Theory

scholarly journals A BETTER COMPARISON OF - AND -COHOMOLOGIES

2019 ◽  
Vol 236 ◽  
pp. 183-213
Author(s):  
SHANE KELLY
Keyword(s):  
Finite Rank ◽  
Valuation Ring ◽  
Direct Proof ◽  
Integral Closure ◽  
Motivic Cohomology ◽  
Valuation Rings

In order to work with non-Nagata rings which are Nagata “up-to-completely-decomposed-universal-homeomorphism,” specifically finite rank Hensel valuation rings, we introduce the notions of pseudo-integral closure, pseudo-normalization, and pseudo-Hensel valuation ring. We use this notion to give a shorter and more direct proof that $H_{\operatorname{cdh}}^{n}(X,F_{\operatorname{cdh}})=H_{l\operatorname{dh}}^{n}(X,F_{l\operatorname{dh}})$ for homotopy sheaves $F$ of modules over the $\mathbb{Z}_{(l)}$-linear motivic Eilenberg–Maclane spectrum. This comparison is an alternative to the first half of the author’s volume Astérisque 391 whose main theorem is a cdh-descent result for Voevodsky motives. The motivating new insight is really accepting that Voevodsky’s motivic cohomology (with $\mathbb{Z}[\frac{1}{p}]$-coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.

scholarly journals About Bredon motivic cohomology of a field

2020 ◽  
Vol 224 (4) ◽  
pp. 106199
Author(s):  
Mircea Voineagu
Keyword(s):  
Motivic Cohomology

scholarly journals Hodge correlators

2019 ◽  
Vol 2019 (748) ◽  
pp. 1-138
Author(s):  
Alexander B. Goncharov
Keyword(s):  
Fundamental Group ◽  
Hodge Structure ◽  
Euler System ◽  
Feynman Integral ◽  
Mixed Hodge Structure ◽  
Modular Curves ◽  
Hodge Structures ◽  
Motivic Cohomology ◽  
Linear Differential ◽  
Mixed Hodge Structures

Abstract Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by twistor plane. We call them twistor connections. In particular, we define the canonical period map on variations of real mixed Hodge structures. We show that the obtained period functions satisfy a simple Maurer–Cartan type non-linear differential equation. Generalizing this, we suggest a DG-enhancement of the subcategory of Saito’s Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin–Selberg integrals. Examples of the motivic correlators include Beilinson’s elements in the motivic cohomology, e.g. the ones delivering the Beilinson–Kato Euler system on modular curves.

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