# Recent developments on noncommutative motives

@article{Tabuada2016RecentDO,
title={Recent developments on noncommutative motives},
journal={arXiv: Algebraic Geometry},
year={2016}
}
This survey covers some of the recent developments on noncommutative motives and their applications. Among other topics, we compute the additive invariants of relative cellular spaces and orbifolds; prove Kontsevich's semi-simplicity conjecture; prove a far-reaching noncommutative generalization of the Weil conjectures; prove Grothendieck's standard conjectures of type C+ and D, Voevodsky's nilpotence conjecture, and Tate's conjecture, in several new cases; embed the (cohomological) Brauer… Expand
12 Citations

#### Paper Mentions

Noncommutative motives in positive characteristic and their applications
Let k be a base field of positive characteristic. Making use of topological periodic cyclic homology, we start by proving that the category of noncommutative numerical motives over k is abelianExpand
A note on Grothendieck’s standard conjectures of type 𝐶⁺ and 𝐷 in positive characteristic
Making use of topological periodic cyclic homology, we extend Grothendieck’s standard conjectures of type C + \mathrm {C}^+ and D \mathrm {D} (with respect toExpand
Noncommutative Weil conjecture
In this article, following an insight of Kontsevich, we extend the famous Weil conjecture (as well as the strong form of the Tate conjecture) from the realm of algebraic geometry to the broad settingExpand
Finite generation of the numerical Grothendieck group
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Grothendieck's standard conjecture of type D in positive characteristic for linear sections of determinantal varieties
Making use of topological periodic cyclic homology, we extend Grothendieck's standard conjecture of type D (with respect to crystalline cohomology theory) from smooth projective schemes to smoothExpand
A note on secondary K-theory II
This note is the sequel to [A note on secondary K-theory. Algebra and Number Theory 10 (2016), no. 4, 887-906]. Making use of the recent theory of noncommutative motives, we prove that the canonicalExpand
Schur-finiteness (and Bass-finiteness) conjecture for quadric fibrations and for families of sextic du Val del Pezzo surfaces.
Let Q -> B be a quadric fibration and T -> B a family of sextic du Val del Pezzo surfaces. Making use of the recent theory of noncommutative mixed motives, we establish a precise relation between theExpand
HPD-invariance of the Tate, Beilinson and Parshin conjectures
We prove that the Tate, Beilinson and Parshin conjectures are invariant under Homological Projective Duality (=HPD). As an application, we obtain a proof of these celebrated conjectures (as well asExpand
HPD-invariance of the Tate conjecture
We prove that the Tate conjecture is invariant under Homological Projective Duality (=HPD). As an application, we prove the Tate conjecture in the new cases of linear sections of determinantalExpand
Noncommutative counterparts of celebrated conjectures
In this survey, written for the proceedings of the conference K-theory in algebra, analysis and topology, Buenos-Aires, Argentina (satellite event of the ICM 2018), we give a rigorous overview of theExpand

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