Quaternionic projective bundle theorem and Gysin triangle in MW-motivic cohomology

AbstractWe reformulate the construction of Kontsevich's completion and use Lawson homology to define many new motivic invariants. We show that the dimensions of subspaces generated by algebraic cycles of the cohomology groups of two K-equivalent varieties are the same, which implies that several conjectures of algebraic cycles are K-statements. We define stringy functions which enable us to ask stringy Grothendieck standard conjecture and stringy Hodge conjecture. We prove a projective bundle theorem in morphic cohomology for trivial bundles over any normal quasi-projective varieties.

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Projective bundle formula for Heller's relative $K_{0}$

Kodai Mathematical Journal ◽

10.2996/kmj/1605063630 ◽

2020 ◽

Vol 43 (3) ◽

pp. 591-602

Author(s):

Vivek Sadhu

Keyword(s):

Projective Bundle

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Motives and modules over motivic cohomology

Comptes Rendus Mathematique ◽

10.1016/j.crma.2006.03.013 ◽

2006 ◽

Vol 342 (10) ◽

pp. 751-754 ◽

Cited By ~ 8

Author(s):

Oliver Röndigs ◽

Paul Arne Østvær

Keyword(s):

Motivic Cohomology

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Néron models and limits of Abel–Jacobi mappings

Compositio Mathematica ◽

10.1112/s0010437x09004400 ◽

2010 ◽

Vol 146 (2) ◽

pp. 288-366 ◽

Cited By ~ 6

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.

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Motivic cohomology of quadrics and the coniveau spectral sequence

Journal of K-theory K-theory and its Applications to Algebra Geometry and Topology ◽

10.1017/is008008012jkt084 ◽

2010 ◽

Vol 6 (3) ◽

pp. 547-589 ◽

Cited By ~ 3

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

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Rost nilpotence and étale motivic cohomology

Advances in Mathematics ◽

10.1016/j.aim.2018.03.027 ◽

2018 ◽

Vol 330 ◽

pp. 420-432 ◽

Cited By ~ 1

Author(s):

Andreas Rosenschon ◽

Anand Sawant

Keyword(s):

Motivic Cohomology

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An Overview of the Proof

The Norm Residue Theorem in Motivic Cohomology ◽

10.23943/princeton/9780691191041.003.0001 ◽

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.

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