The decomposable part of motivic cohomology and bijectivity of the norm residue homomorphism

1992 ◽  
pp. 79-87 ◽  
Author(s):  
Bruno Kahn
Keyword(s):  
Motivic Cohomology ◽  
Norm Residue

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.

The Norm Residue Theorem in Motivic Cohomology

2019 ◽  
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Residue Theorem ◽  
Motivic Cohomology ◽  
Norm Residue

The Norm Residue Theorem in Motivic Cohomology

2019 ◽  
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Residue Theorem ◽  
Motivic Cohomology ◽  
Norm Residue

The Norm Residue Theorem in Motivic Cohomology

2019 ◽  
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Large Scale ◽  
Residue Theorem ◽  
Geometric Constructions ◽  
Complete Proof ◽  
Motivic Cohomology ◽  
Symmetric Powers ◽  
Norm Residue ◽  
Cohomology Operations ◽  
Many Sources ◽  
First Time

This book presents the complete proof of the Bloch–Kato conjecture and several related conjectures of Beilinson and Lichtenbaum in algebraic geometry. Brought together here for the first time, these conjectures describe the structure of étale cohomology and its relation to motivic cohomology and Chow groups. Although the proof relies on the work of several people, it is credited primarily to Vladimir Voevodsky. The book draws on a multitude of published and unpublished sources to explain the large-scale structure of Voevodsky's proof and introduces the key figures behind its development. It proceeds to describe the highly innovative geometric constructions of Markus Rost, including the construction of norm varieties, which play a crucial role in the proof. It then addresses symmetric powers of motives and motivic cohomology operations. The book unites various components of the proof that until now were scattered across many sources of varying accessibility, often with differing hypotheses, definitions, and language.

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

scholarly journals Rost nilpotence and étale motivic cohomology

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