scholarly journals Noncommutative motives, numerical equivalence, and semi-simplicity

2014 ◽  
Vol 136 (1) ◽  
pp. 59-75 ◽  
Author(s):  
Matilde Marcolli ◽  
Gonçalo Tabuada
Keyword(s):  
Numerical Equivalence ◽  
Noncommutative Motives

scholarly journals Kontsevich’s noncommutative numerical motives

2012 ◽  
Vol 148 (6) ◽  
pp. 1811-1820 ◽  
Author(s):  
Matilde Marcolli ◽  
Gonçalo Tabuada
Keyword(s):  
Numerical Equivalence ◽  
The One ◽  
Noncommutative Motives

AbstractIn this article we prove that Kontsevich’s category NCnum(k)F of noncommutative numerical motives is equivalent to the one constructed by the authors in [Marcolli and Tabuada, Noncommutative motives, numerical equivalence, and semisimplicity, Amer. J. Math., to appear, available at arXiv:1105.2950]. As a consequence, we conclude that NCnum(k)F is abelian semi-simple as conjectured by Kontsevich.

Matching, Counting, and Conservation of Numerical Equivalence

1983 ◽  
Vol 54 (1) ◽  
pp. 91 ◽  
Author(s):  
Karen C. Fuson ◽  
Walter G. Secada ◽  
James W. Hall
Keyword(s):  
Numerical Equivalence

scholarly journals Hochster's theta pairing and numerical equivalence

2014 ◽  
Vol 14 (3) ◽  
pp. 495-525 ◽  
Author(s):  
Hailong Dao ◽  
Kazuhiko Kurano
Keyword(s):  
Grothendieck Group ◽  
Three Dimensional ◽  
Hypersurface Singularity ◽  
Isolated Singularity ◽  
Divisor Class Group ◽  
Intersection Multiplicity ◽  
Counter Example ◽  
Mild Assumption ◽  
Numerical Equivalence ◽  
Torsion Free

AbstractLet (A, ) be a local hypersurface with an isolated singularity. We show that Hochster's theta pairing θA vanishes on elements that are numerically equivalent to zero in the Grothendieck group of A under the mild assumption that Spec A admits a resolution of singularities. This extends a result by Celikbas-Walker. We also prove that when dimA = 3, Hochster's theta pairing is positive semi-definite. These results combine to show that the counter-example of Dutta-Hochster-McLaughlin to the general vanishing of Serre's intersection multiplicity exists for any three dimensional isolated hypersurface singularity that is not a UFD and has a desingularization. We also show that, if A is three dimensional isolated hypersurface singularity that has a desingularization, the divisor class group is finitely generated torsion-free. Our method involves showing that θA gives a bivariant class for the morphism Spec (A/) → Spec A.

scholarly journals Computing Néron–Severi groups and cycle class groups

2015 ◽  
Vol 151 (4) ◽  
pp. 713-734 ◽  
Author(s):  
Bjorn Poonen ◽  
Damiano Testa ◽  
Ronald van Luijk
Keyword(s):  
Equivalence Classes ◽  
Class Groups ◽  
Tate Conjecture ◽  
Etale Cohomology ◽  
Numerical Equivalence ◽  
Étale Cohomology ◽  
Integral Variety ◽  
Cycle Class

Assuming the Tate conjecture and the computability of étale cohomology with finite coefficients, we give an algorithm that computes the Néron–Severi group of any smooth projective geometrically integral variety, and also the rank of the group of numerical equivalence classes of codimension $p$ cycles for any $p$.

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